Graduate School (Can I be a Mathematician?) I am posing this question for myself, but many people surely are wondering the same, so I think this is valuable for others as well as myself. There are similar questions on the site but they do not ask exactly what I'd like to know. Let me share my own experience to illustrate what I am wondering.
I am currently transitioning to upper division math course work. I have earned all A's in my lower division math classes and some upper division ones at a relatively difficult university, but most times feel as though I just barely pulled off the grade, and only through copious studying. Even when I get near perfect marks, I feel I hardly understand the material. I also tend to forget material quickly. I look at questions from calculus, differential equations, and linear algebra for instance that I only recently learned (math GRE or qualifying exam type questions) and feel I should certainly know the answer seeing as I've taken the course, but can't get to it.
I love mathematics, and spend just about all of my time doing it. The problem is, I see others who are quicker, see deeper, need less examples, and so on, when it comes to understanding the material of any given math class. I want to be a professional mathematician (a professor, researcher, or similar), but I honestly question if I am being naive with this career objective. I wonder if hard work truly can get me where I want to go, or whether I am not cut out for this. 
Am I naive thinking I can make a career for myself as an academic mathematician, when there are people years younger who surpass me with their skills and experience? This is a difficult question to pose, because for the last two years my life has been dedicated to mathematics. I would like to dedicate more if not all of my life to math, and go to graduate school for the subject and so forth, but not in vain.
Any honest input would surely help me and likely many others who are in similar situations. 
Thank you
 A: I was in a similar situation to yours when I first started falling in love with Mathematics at 18. I had a deep passion for the subject, did well on exams when I put in the time, and spent many days and nights studying and learning intensely. I loved it for its beauty and difficulty, gaining confidence when I could learn and solve difficult problems. As I progressed through my undergraduate studies, I of course noticed others who seem to have a "natural" or gifted ability in it. Merely studying for a day in advanced would earn them just as high and if not higher grades than I when I have put blood and tears to earn my grade.
I would sometimes be frustrated by this, comparing myself to individuals who could party on the weekends study for a day or two and casually get an A on what I thought were tough exams. I shared my feelings to close friends and I learned that you cannot compare others to yourself in that way. Everyone comes from different backgrounds and experiences they have had throughout their lives that led them to be in the same class and take the same exams as you did. Some developed a love and curiosity for math at earlier ages perhaps and have had much longer training in solving problems and thinking about mathematics. Before I had a passion, I liked surfing, snowboarding, arts, chemistry, an array of topics that are not just mathematics. Naturally, some of  those who have practiced more are going to tend to be better.
That being said, going to class, studying, and getting A's on exams is not what is going to make you a good mathematician. Of course understanding the foundation is essential, but to become a researcher or professor you must love and enjoy doing research. 
During the last semester of my undergraduate I got involved in a research topic with a probability theory professor. The topic involved developing a percolation model to model "pores" (spaces between plates) and develop a probability function that would return a measure of the likelihood plates would slide, causing an earthquake. This was my first research experience and although difficult to do gave me a sense of what it would be like as a graduate student in a PhD program. I was hooked to say the least.
I wanted more, so I searched for more undergrad research opportunities and eventually got involved with UCLA's REU program. There I did research in image processing and microscopy. From there on I really fell in love with doing research and starting studying and researching PhD programs in mathematics. 
I also found that studying for the GRE would not be that difficult, with the assumption that doing well in higher math classes I would also do well on these standardized tests no problem. Again, I had the wrong assumption. Taking timed standardized tests are very different then taking exams in college or doing research. They require a different kind of training and also need time and preparation to do well in them. I personally had to take the GRE three times before I felt like I scored high enough to submit my score to various schools I was applying for in graduate school.  
To conclude and halt my banter, I did get into a competitive PhD program and still 10 years later I am in love with mathematics, learning, and research. Thus to conclude I would offer this advice; find undergrad research opportunities to gain some experience, start looking into pure math (e.g. real analysis, abstract algebra, etc.) if you haven't already, learn a programming language (Python, C++, etc.), believe in yourself ("I can, I must, I will"), and enjoy the journey. Any journey that is not hard wont be fun anyways :).
A: "Hell of a semi-emotional semi-rambling very-philosophical-and-anecdotal wall-of-text by an amateurish undergrad" warning.


"I love mathematics, and spend just about all of my time doing it. The problem is, I see others who are quicker, see deeper, need less examples, and so on, when it comes to understanding the material of any given math class."

I should communicate a hard lesson I had to learn - and one I am indeed still learning in my own academic career thus far would be one my complex analysis professor touched on and I eventually elaborated on knowing my own experiences with people. It's still a bit hard for me to grasp and my own self-esteem and self-doubt issues don't help this situation at all whatsoever ...
Let me set the stage a bit. I came out of that class after the midterm a few months back feeling pretty low and I didn't think I had any hope whatsoever. I've been having a rough semester in terms of my mood - getting stuck on stuff, stuff of that sort. Stuff that seems so basic that it hurt me at the core of my being. Here I am, math being pretty much the one thing I'm good at, and I'm feeling like I'm not even good at that now. It's not for a lack of doing good on the midterm (I got like ... a B+ or something?), it's just that I felt like I was falling short. Especially seeing people getting straight A+'s, doing well on any bonus problems, talking constantly in class (I'm nearly mute from social anxiety) ... it hurts. That my professor for this class is pretty much renowned as the best math professor in the university and was the previous head of the department didn't help matters because I felt like I was wasting this man's time, even worse because he might get some input as to whether I get into grad school.
So yeah ... it's a complex issue and it's not the best analogue for your situation, but I'm getting to the point.
What this professor said was that I was one of the top students in his eyes (and, after the final last Thursday at the time of writing, said I was the top student).
"What? Why?" That was easily the first thought to come to mind. Here are people making straight A's, and here I am struggling to float above that 90% line to an A (I really really want an A because I need to raise my GPA some for the uni's math program). Me, making dumb mistakes on the homework and, when I say something in class, it's probably dumb?
It was because it was a sign that I was thinking through the material. Sure, I was making mistakes - and making even more mistakes than I was turning in, but I at least had to skill to realize where I was screwing up. But that was a sign I was thinking things through, rather than slapping something down - right or wrong.
I came to realize, as I thought over the conversation we had, that being good is, for one, not simply a matter of grades. Grades don't tell the full story. Some people are truly gifted and can get everything right off the bat; you could see them almost as a reincarnation of Euler or Gauss. Others have to study hard and make a hell of a lot of effort and lose a lot of sleep to scrape for that A.
Being good is neither of these alone, but a mix. What makes a person good at mathematics? Is it being a literal sponge, able to soak in ideas and know and understand them? It certainly helps but it doesn't tell the whole story. I think more that math, in itself, is also about the underlying thinking. Math, at its very core, is about logic. Thought processing. Reasoning. Ideas and theorems. Piecing all these things together to form more ideas, more theorems, formulas, proofs, algorithms, applications, all sorts of things.
Does being slow necessarily inhibit you to do any of these things? So maybe someone can finish a test half an hour or more faster than you, or they have the homework done before class has even ended and here you are, the night before it's due, it's five in the goddamned morning and you usually get up at seven, regretting every life decision that got you to this sleep deficit. (Or worse, like me, you get stuck and have nightmares on the problems you get stuck on ... I wish I could say I was joking, I can get OBSESSED.)
Is everyone going to always be right? I guarantee that, at some point or another, Euler or Gauss - people renowned as the greatest mathematicians in history - were wrong or made a mistake. This isn't some "Einstein got an F in math" bullcrap. Being wrong is important too! Even moreso the ability to know you're wrong or have no idea what's going on! It's sounding dumb, isn't it? "Knowing you're wrong is important? Isn't the whole point to be right?" Well, it's not so much knowing you're wrong, as much as knowing why you're wrong. Learn from your mistakes. Tripping is an important part of the learning process for this reason - because knowing why you're wrong let's you understand any misconceptions you might have, and thus prevent that mistake from happening again. Find yourself in a class and at some point you just have no goddamned clue what the hell is going on? Realizing that is important - you know, then, you need to put in more effort, or ask for help.
Imagine not knowing you're wrong and just turning in a paper full of mistakes. Or sitting there, knowing you know jack about what's going on, only to remain silent so you don't look or feel stupid. I guarantee you'd feel more stupid the moment you get 20 out of 100 points on a test or something because, here you are, having been lost for two months. The ability to realize you need help is critical in learning. That you know you're wrong is important - if for no other reason that you're right that you're wrong. You're right about something if you frame it that way lol.
But even more importantly: being wrong also helps hone your instincts for when you're wrong. At some point you just develop this sixth sense that's, like, you know something's up. You might be able to pin it down, you might not, but even that sense is important. It can be a key for further investigation and analysis, or to seek help. If nothing else, honing that sense also hones the same sense for when you're right, you know? The two go hand-in-hand like that, weirdly. Being always correct never hones you're understanding of being wrong, but being wrong helps you on both.
Phew.
"What's important in mathematics, for success?" is probably the better question. "Talent? Effort?"
I feel like it's a blend. To some degree, there's a talent. Some people just naturally have the inclination to "think mathematically." To some, there's effort. Put your mind to anything and you can manage. (I remember one day thinking I'd never understand calculus.)
But everything in moderation.
The person who has a lot of talent and puts in no effort? They're probably those people that are right about everything, know everything, could probably make the professor's lessons more efficient if they wanted to (albeit at the cost of everyone's own learning experience). The problem? They don't know when they're wrong. Like I said earlier - the ability to know you're wrong is important in addressing your misconceptions and avoiding future screw ups.
The person with a lot of effort and no talent? Well ... they might have some ability in math, there's no doubting that. They take part in an uphill battle, however. Everyone should play to their strengths and talents at least somewhat. This touches a bit on my feelings about what a commenter noted: that one is going to be inclined to be good at what they love.
Is there something other than math you love? Would you say you're at least decent at it? I love playing video games. I'm probably one of the most knowledgeable people on Pokemon short of, maybe, hardcore speedrunners and world-class competitive players. If I get on a good day, I can speedrun Super Mario Bros. in 5-6 minutes. Sure, I never even got to regionals in Pokemon VGC, and the world record for Super Mario Bros. speedrunning is like 4:55 and requires optimizations requiring 1/60-th of a second reflexes, but to say I'm better than most people isn't a stretch. I've grown up with these games, you know? Of course I'm good!
In that sense, passion and talent sort of go hand in hand, even if why is not immediately apparent. (Conversely, it might also mean that the person without seemingly natural talent also isn't truly passionate about what they do. Depending on your personal outlook, that's actually worse than anything here - you might be a damn genius at whatever your talentless effort brings you to, but doing something you don't like for so long is soulcrushing, isn't it?)
You've made it to, what, the calculus and linear algebra series of mathematics so far? I have a bunch of friends with whom I tend to show such levels of things and they're just astounded that I can understand what's going on. Most people don't really need to go beyond algebra for their daily lives - and most certainly wouldn't. Being good thus far - and loving it - is certainly a good sign of a future in mathematics for you, even if it came about as a result of effort and you weren't perfect. (No one's perfect, I didn't have a good grasp of linear algebra until after like 4 courses that heavily involved it in some way or another.)
That said we also should touch on the two "kinds" of math. I feel like there are two main kinds of mathematics - the "computational" and the "logical" kind. Computational covers up until your sophomore/junior level classes in college - basic algebra, calculus, linear algebra, etc. In these classes, you don't do too many proofs, you focus on learning kinds of mathematics, properties of it, formulas, algorithms, problem-solving techniques, etc. In the "logical" kind, you focus on proofs - actually showing where these identities, ideas, theorems, whatever, come from, and laying the groundwork for you to do the same.
You could be good at one kind and not another. Or maybe you're good in both. Having a balance might be best for a career in mathematics specifically; the computational kind isn't bad and there are some fields that are much more computationally heavy, and there are also tangentially-related STEM fields that you could get some sort of foundation in even if not strictly math. (STEM is nice in that there's a lot of blending together with math. Academic talent in mathematics certainly sets a nice groundwork for a lot of places in STEM.)
I think it might be important to be aware that you might be in for a hell of a ride once you get into your more "logical" type courses - you're on the verge of them, it sounds like. It's possible you might not find them to your liking. Or who knows, depending on your courses thus far, you might already be familiar with such - I had a friend whose high school calculus class was at the same level of rigor as my real analysis course. It's crazy out there.
So, in summary:


*

*You're about to hit a big transition point for "computational" style math to "logical" style. You could be good at one or the other or both; being poor at one doesn't mean you suck at either. Play to your strengths in your career.

*Success in mathematics is a mix of talent and effort. A mix of both is great, and too much of either sets yourself up for troubles down the road in some respect or another.

*Being wrong is okay. Knowing when you're wrong is better. What's important is that you're thinking, and learning. Being wrong is a natural part of that process, and to expect perfection out of yourself would be absurd. You don't need to make straight A's.

*Passion is always important, and often goes hand in hand with talent, even if said talent is not immediately clear. Like I said, there's already one big transition point in your mathematical career staring you right in the face, even if you don't know it yet. And mathematics is a broad, wonderful field of study - there's almost certainly some aspect you're talented in if you love it, even if you don't realize it yet, because you might not have found it. 

*Play to your passions and strengths. Don't be afraid to seek help - ever. Understand that it's important to be wrong, just as it is important to be right - contextualize this as being right about being wrong if nothing else, as the ability to destroy one's own flawed arguments is a good ability. If you truly love math, and are putting in genuine effort as you seem to be, I feel like you'll probably have a good chance of success in math.
Phew.
Okay, sorry for this insane ramble. You just touched on some of my own insecurities which I've been dealing with myself. To anyone insane enough to read all this (and hopefully find it useful however anecdotal my experiences are), here's a metaphorical cookie.
