Should I be worried that I am doing well in analysis and not well in algebra? I attend a mostly liberal arts focused university, in which I was able to test out of an "Introduction to Proofs" class and directly into "Advanced Calculus 1" (Introductory Analysis I) and I loved it. I did great in the class. I was not very mathematically mature at the time, but I studied hard and started to outpace many of the senior level students who had a least a good year or more of experience than me. Furthermore, the professor teaching the course was apparently known to be particularly difficult, but I loved his course. I enjoyed the challenge and wound up with a B+, the 2nd highest grade given in the class. I took Advanced Calculus 2 and loved it even more. The professor even suggested that I take a graduate complex analysis course in the Fall. (Just a side note here, the undergraduate complex analysis course at my school does not use any theorem's or proofs. The grad version is similar to say, an honors undergraduate course at more traditional math program.) I took this as a high complement, and a verification that I was in fact doing well. I know I am not very deep into analysis, but I feel comfortable with the subject, even with the more abstract parts.
However, I am really struggling with abstract algebra. I can't understand why. I study the material really hard. I am doing better than most in the class, and I am maintaining a solid B average, but I really have trouble thinking about algebra like I do analysis. I feel like I am mostly just regurgitating theorems and techniques just to pass the exams. I know I can pass the course, but I also know that this mindless memorization will eventually come back to haunt me later on in my mathematical career. Algebra is truly one of the pillars of math which is why I really feel terrible that I don't understand it.
Is this a sign that I simply don't have what it takes to succeed in math? I would love to go on to graduate school and hopefully get a PhD. In fact, a professor actually said to me, "I think it would be a shame if you didn't go to grad school for math." He told me that before I took algebra, but now I feel like my world is "crashing down" in a sense. Before I was a "good" student; now, I feel like a zombie in the back of the room. Any input is greatly appreciated, but what I really want to know is, has this happened to anyone who has gone on to succeed in a Ph.D math program?
 A: I believe that I may be of some consolation. 
I had a very similar experience to you. I started doing "serious" math when I was a senior in high school. I thought I was very smart because I was studying what I thought was advanced analysis--baby Rudin. My ego took a hit when I reached college and realized that while I had a knack for analysis and point-set topology, I could not get this algebra thing down! I just didn't understand what all these sets and maps had to do with anything. I didn't understand why they were useful, and even when I finally did grasp a concept I was entirely impotent when it came to those numbered terrors at the end of chapters.
I held the same fear that you do. I convinced myself that I was destined to be an analyst--I even went as far to say that I "hated" algebra (obnoxious, I know). After about a year of so, with the osmotic effect of being in algebra related classes, and studying tangentially related subjects, I started to understand, and really pick up on algebra. Two years after that (now) I would firmly place myself on the algebraic side of the bridge (if there is such a thing), even though I still enjoy me some analysis!
I think the key for me was picking up the goals and methods of algebra. It is much easier for a gifted math student to "get" analysis straight out of high-school, you have been secretly doing it for years. For the first half of Rudin while I "got it", this was largely thanks to the ability to rely on my calculus background to get why and how we roughly approached things. There was no such helpful intuition for algebra. It was the first type of math I seriously attempted to learn that was "structural", which was qualitative vs. quantitative. My  analytic (read calculus) mind was not able to understand why it would ever be obvious to pass from ring X to its quotient, nor why we care that every finitely generated abelian group is a finite product of cyclic groups. I just didn't understand. 
But, as I said, as I progressed through more and more courses, learned more and more algebra and related subjects, things just started to click. I not only was able to understand the technical reasons why an exact sequence split, but I understood what this really means intuitively. I started forcing myself to start phrasing other parts of mathematics algebraically, to help my understanding. 
The last thing I will say to you, is that you should be scared and worried. I can't tell you how many times in my mathematical schooling I was terrified of a subject. I always thought that I would never understand Subject X or that Concept Y was just beyond me. I can tell you, with the utmost sincerity, that those subjects I was once mortified by, are the subjects I know best. The key is to take your fear that you can't do it, that algebra is just "not your thing", and own it. Be intrigued by this subject you can't understand, read everything you can about it, talk to those who are now good at the subject (even though many of them may have had similar issues), and sooner than you know, by sheer force of will you will find yourself studying topics whose name would make  you-right-now die of fright. Stay strong friend, you can do it.
A: I am also an undergrad student. I suggest you to calm down. I know I will do a math PhD, so I work hard to enjoy the math and feel no pressure. Sometimes or often I have difficulty in some areas that might be trivial to many people, but I don't feel the rush because I know one day I will understand them, simply by going through them over and over again.
I often compare studying math to playing a computer game with infinitely many levels that gets exponentially harder. Sure, it's good if you make lots of progress and go through many levels fast, but sometimes the point of playing the game is to enjoy the moment and not to always aim at next level while you are playing your current level, because sooner or later you will get stuck at some level and might not go to next for a long long time.
So relax and study and enjoy.
A: Some people are just naturally more analytic than algebraic, and vice versa.  Personally, I do research level algebra, but if I see $\epsilon$ and $\delta$ on the same page I run screaming.
That's not good though, so I'm making myself do it.  I enrolled in a complex analysis class, and awful as y'all's side of the fence is, I'm sticking to it.  And though I'm not doing the best in the class, y'know, I'm starting to enjoy parts of it.
You can't expect to excel in every area as a mathematician, so focus on rocking at the stuff you do like to do, but make sure your head stays above water in the areas you don't like.  You never know when you might end up needing algebra to accomplish something in analysis.  If you need motivation, try thinking about hybrid disciplines like functional analysis / operator theory (there's even an "algebraic analysis").  Relate theorems you learn in algebra back to analysis in any way you can.  It will help you remember them and maybe give you some cool ideas for later.
A: No reason to be alarmed or worried...it's too early for you to be in a position to worry about it. For a first crack at abstract algebra, don't fret. Most undergraduate math majors inevitably do fall into "analyis-oriented" and "algebra-oriented", just as in highschool, there is often a "partition" of students into those who prefer high school algebra and those who prefer geometry.
But, it takes more than one course to know this about yourself. Abstract algebra, when I took it as an undergraduate, was typically the gateway course into higher-level abstract math. Personally, I loved it! But there were also classes I developed a love for after covering the "tools" and language of the field: i.e., only after having taken a class or two. 
If it's any consolation, you are encountering abstract algebra at a very young age, and though you no doubt have mathematical talent, it does take "time" and effort, and not just raw talent, to develop the cognitive and mathematical maturity to reason abstractly and to develop a sense of "grasping a subject intuitively." Sometimes this is facilitate by classes like an "Introduction to the language and practice of mathematics" (which a Univ near me offers as a bridge between calculus/differential equations and introductory linear algebra, and all subsequent course offerings. 
At any rate, wrt becoming comfortable with the more abstract nature of what you're encountering: It is largely a matter of exposure to and engagement with abstract math to operate comfortably within that realm, but there is also a purely developmental component which impacts the ease with which this "acclimation" occurs.   


Is this a sign that I simply don't have what it takes to succeed in math?

No, it is not. Every math student I know, sooner or later, has encountered a point where they ask themselves that very question. Often times, more than once. 
How you respond at points like this, and how you respond when you feel overwhelmed or intimidated (and you will feel that way again, if you persist in your studies!):  that will determine whether you have what it takes to succeed in math.
A: All math are abstract. All are equally difficult at higher level. All takes enough devotion in order to succeed. Have faith in yourself. Intelligent and ingenious steps result only through hard and continued effort. First try to finish cover to cover "Jacobson's vol.I, Abstract Algebra". This book will definitely enhance your taste for AA.
A: tl;dr: just find another area of math, there's too many to sweat this. But also consider taking number theory to really whet your interest in algebra before you dismiss it.
Firstly, I just really think you might be taking the wrong approach on what "math" means. In some circles math means the purest math possible with the most abstract rigorous proofs and the most challenging problems. Think: number theory, commutative algebra, algebraic geometry, differential geometry. Generally, algebra fields. Try not to let those circles determine your self-worth as a mathematician, and believe me, it's a strong effect.
It's better to find a lovely mathematical science that's a wonderful fit. You'll need math in all of them, and some utilize more math theory than other. This sounds silly but I'm reading Scott Page's Complex Adaptive Systems book right now and it contains a surprisingly pleasant and cogent primer on how discourse, thought experiments, mathematical theory, and computational theory play differing roles in different scientific fields. For instance, economics used to be dominated by discourse (think Adam Smith), then sometime in the 70s or 80s the mathematicians essentially wrested the field and many argue to the detriment of all other modes of thinking since inductive evidence was basically banished, and recently computational models have taken storm more.
You could pick a math with any of these tools focuses, including many within math, ranging from abstract, algebraic proofs, to geometry-flavored proofs, to computation-flavored deductions!
Secondly, it might be the class, not the subject. Why not try using algebra to solve an interesting problem! Number theory is ripe with these. Try using your algebra to solve such facts as this, in very rough order of most basic/too obvious to be interesting to deep:


*

*Prove that for any natural number $a$, $a*0=0$.

*Use ordering properties of the integers to prove that if $a$ divides $b$, then $|a| \leq |b|$. 

*Use the well-ordering principle to prove there are no integers between $0$ and $1$.


That's the warm-up!


*

*Prove Fermat's Little Theorem.

*Generalize it to Euler's Theorem: $a^{\phi(m)} \cong 1 (\mod m)$.

*Prove the Chinese Remainder Theorem.

*Prove that any element $a$ has a well-defined order $n$ such that $a^n \cong 1 (\mod m)$ and $n$ is smallest. Now prove $n$ divides all other numbers $m$ such that $a^m \cong 1 (\mod m)$.

*Say $a$ is a quadratic residue modulo $m$ if it has a square root $n$ such that $n^2 \cong a (\mod m)$. Can you find a formula to determine if $-1$ is a perfect square $\mod m$ if $m$ is prime?


These vary from algebraic in nature to more number theoretic, but the building blocks are there. (Hint: Fermat's Little Theorem is a special case of Lagrange's Theorem of group theory). I think it's far more fruitful to learn many important algebra concepts over the ring of integers and the groups $\mathbb{Z}_m$ over addition, and the units in that wrong over multiplication. Then manipulating abstract symbols gets more fun as you consider polynomials and the wonderful things they can tell you about the actual integers.
A: I heard somewhere that "Abstract Algebra is the class that separates the boys from the men", so I'd be worried... algebra is more abstract, with concepts that are hard to impossible to visualize. And mathematics is mostly about abstractions. You need to get acquainted with them.
Just my 2¢ as a mathematics minor...
Good luck! Just keep at it, get other sources for different viewpoints/teaching technique/emphasis, not everybody learns the same way. Try doing exercises, check out what cooks here in the area, answer questions.
