undergraduate math vs graduate math It's really a mild, soft question.
So far, I am an undergrad student, contemplating on several majors.
What will be the major difference if I become a math major and go to a graduate school to study math?
Will a graduate math major generally focus on reading/studying recent research papers? Or will  the graduate math major be taught by professors on the things not covered in undergrad programs? 
The reason why I am asking this question is that undergrad math programs seem to me at this point so well-covered that after studying the programs, a person will be able to do anything he wants to do, and if one wants to do research, he will be able to catch up with recent progress by reading research papers.
If this is true, why is the graduate school even needed?
By the way, I am a first-year undergraduate student :)
 A: From one who failed to realize all the value of his own education... and to provide a wider perspective to @MattPressland's excellent answer.
Although written for high school students, I believe that What You'll Wish You'd Known by Paul Graham resonates deeply with your question. Specifically in wondering "why is the graduate school even needed", I think the following quote sums the essay and answers your question:

Suppose you're a college freshman deciding whether to major in math or economics. Well, math will give you more options: you can go into almost any field from math. If you major in math it will be easy to get into grad school in economics, but if you major in economics it will be hard to get into grad school in math.

This applies to graduate school in that having a Masters in Mathematics will provide you with more options than a bachelors alone. Although theory is often not as applicable in practice, I'd certainly hire the PHD who could show the output of their labor over the BA who could demonstrate the same.
A: Here is what I tell my grad students:
The difference between undergrad mathematics and graduate mathematics is the difference between art history, or art appreciation, and learning to be an artist.
As an undergraduate you see a lot of mathematics, but you don't create new mathematics.  The goal of graduate school (and here I am speaking from experience with top fifty U.S. graduate schools, so what I am saying probably applies best in that context) is to learn how to create new mathematics, and then to create that new mathematics.   
One specific consequence of this (in my view) is the following: often in undergraduate mathematics classes, proofs and rigor are presented almost as moral imperatives --- as if it is a moral failing to know a statement without knowing why it is true; consequently, people often put a lot of effort into learning arguments just for the sake of having learnt them.   (This is exaggerated, perhaps, but I think it reflects something real.)  On the other hand, in research, one learns arguments for different reasons: to learn technique, to pick out important ideas --- there is a professional aspect to the way one looks at pieces of mathematics which is not usually present in undergraduate mathematics.  One gives proofs in order to be sure that one hasn't blundered; one's interaction with the mathematics and the arguments is much more visceral than in undergraduate courses.   
(I am not speaking from any experience now, but I think of the difference between learning how to interact with a block of marble, and bring a new form out of it, however rough it might be, in comparison to looking and learning about a lot of existing beautiful statues, masterpieces that they are.)
A: As a graduate student, the most useful skill I learnt as an undergraduate was not the mathematics itself, but how to learn mathematics. The edge of the subject is so wide that it's mostly not practical to get to a lot of current research problems as an undergraduate, even in a particular subfield like geometry or algebra for example.
That's not to say that the mathematics isn't important (and in fact I'm probably underplaying its importance because the parts of it I use all the time have become second nature), but knowing how to learn things efficiently is incredibly useful.
The graduate school experience probably varies quite a lot from university to university (and between countries as well), but my experience is similar to that described by Asaf in the comments - you still do some more formal courses at the start, but more independently than as an undergraduate, and at the same time your supervisor will suggest things you should read and problems you should think about - and these should lead to you discovering more things to read and problems to think about under your own volition.
I should probably also recount what I've always heard said by lecturers as the big difference between begin an undergrad and a grad student - as a grad student, you have to contribute original research. The upshot of this is that while the problems you see as an undergraduate may be difficult, they at least have answers, but this need not remain true when you are a grad student, and learning how to make judgements about which questions are worth persuing is an important aspect of postgraduate study - as mentioned by Eugene, Terence Tao has lots of good advice along these lines.
This also leads to a sound-bite answer to your question "why is graduate school even needed?" - because the process of learning how to do research is distinct from the process of learning how to do mathematics.
A: Given that you are just starting as an undergrad in Math, it is highly unlikely that you understand yet what math is.  What you have (likely) seen so far probably peaks with calculus.  What that means is that you have learned how to apply methods without having learned how to think mathematically and how to prove the theory that underlies subjects like calculus.
It is really not until you take a course in number theory, or a very theoretical initial coverage of linear algebra, that you are exposed to proof methods.  (My recommendation, btw, is to take an introductory course in logic as soon as possible.  Best course I ever took on my way to a B.S. and eventually a Ph.D. in mathematics.)  You may find out that Math is just not for you.  
Beyond that, many other subjects that you may be contemplating as a major will require considerable math.  So, take as much as you can.  I always recommend people double major, with one major being math.  Key to success in graduate school in any science or engineering subject (or economics or ...) is mathematics.   
Beyond this, it pays to be somewhat practical.  It was discussed that there are more opportunities in Statistics.  You could also consider Applied Mathematics.  Both can be as mathematical as pure mathematics, and the job opportunities (and expected salary) are much better.  Some of the best mathematicians I ever encountered with applied mathematicians and engineers.  
Finally, realize that computer science is really just applied mathematics, especially if you go on to graduate school.
Whatever you  do, pursue a subject that you enjoy.  Don't do it because your parents expect you to.
A: A short, but I hope informative, story from someone with a PhD in Physics, but who also was a professor of Mathematics for a few years.
I was teaching in the physics department at one of America's finest colleges and worked with a brilliant undergraduate math major and published three journal articles with him.  He got straight As, and seemed to be able to solve any problem I gave him... including one that I probably could have solved, but it would take me longer than he did.  He had such a bright future.
He was accepted to math graduate school at one of America's very top math departments but alas didn't thrive there.  There are a few fundamental difference between undergraduate education and graduate education, which he couldn't navigate.  (He left school to program computers for a bank.)  Speaking in broad terms:  


*

*Undergraduate:  You learn math that is already understood by experts, and you solve problems posed by others (professors or books), and are under a fairly strict schedule of classes, exams, paper deadlines, etc.

*Graduate:  You create new math that is fully understood by nobody else (yet), you must search and create your own problem, and have very few scheduling commitments, so you must be self-motivated.


This is not to dissuade anyone from attending math graduate school... quite the contrary.  I want to alert such prospective students so they know what to expect.  Actually, the best preparation at the undergraduate level is to do true research (appropriate to your skill and education) which involves creating your own math problem, and then doing your best to solve it.  
One of my criticisms of many undergraduate departments (not just in math or physics) is that they don't teach and encourage students to make their own problems.
So my advice to an undergraduate interested in math graduate school would be to meet with your math advisor and try to arrange a self study (or "independent topics" or whatever is the format at your school) and "test the waters."  You may love it (!) in which case graduate school sounds right for you.  If not, I think you'll have to decide whether you want to develop such skills (in preparation for graduate school) or instead pursue a career where others will give you problems to solve.  There is no shame whatsoever in such a career.  
A fun little diversion, which nevertheless illustrates the difference between undergraduate and graduate mathematics education.  Consider your favorite discipline in math (number theory, geometry, differential equations, ...) and just start asking questions/problems in that discipline.  don't worry that you cannot solve them!  (Fermat couldn't solve his Last Theorem, despite what he wrote in his margin.)  Just keep asking and asking them.  If you find you're stuck, or you can only come up with problems you've already seen, or if you're frustrated by such an effort, stop and think about that.  If instead you get energized, excited, and motivated by this little exercise--especially if you come up with a "gem" of a problem--then you may have the temperament and skills for graduate research.  (Of course this isn't the only requirement for success.) [You can search for my TEDx talk on "how to ask good questions" where I run through this exercise for sudoku puzzles.]
You want to follow a career path that gives you the greatest personal satisfaction and helps the discipline and society more broadly.
Good luck!
A: Terence Tao has great advice for mathematicians at every stage of their careers.
http://terrytao.wordpress.com/career-advice/
A: To get an idea of how mathematics graduate school in the United States works I would suggest the book A Mathematicians Survival Guide by Steven Krantz. The books is filled with information about the ins and outs of mathematics graduate school, including advice concerning many of the common pitfalls experienced by graduate students. It also contains advice for recent PhDs about their options after graduate. Overall, a very useful source for those unsure of whether graduate school is for them. The preview on Amazon should give you a good idea about the contents of the book. 
A: User2346:
Here's the opinion of someone who majored in math (at a top school in the United States), but was not a genius at it. 
This is a condensed, generalized, down-and-dirty opinion, but I stand by it.
Your assumption about being well-covered is more accurate that not. If your intent is to form a mathematical basis of strength that you can apply to applications of math (e.g. finance stat econ) then undergrad is as far as you need to go with it. By the time you take Real Analysis & Complex analysis your mind will have developed in an amazing way. As Matt Pressland said, you learn how to learn Mathematics. Matt also mention the difference between that style of meta-cognition and being able to research. It's impossible to say today that you could or couldn't come up with a unique research idea that would be considered a contribution and that you're passionate about to pursue. However, unless the word research makes your heart throb, you don't need Math graduate school.
I've rewritten the rest of my answer to be a bit more compact.


*

*Do not major in math, minor in it. You will gain a superior analytic mind. However, your employment potential depends on not being a Math major. The Corporate world (Actuarial aside) doesn't know wtf Math is. They will ask if you wanted to Teach High School.

*Accounting is boring, and Economics is fluffy. Choose Statistics and augment with Finance. Also, Computer Science requires more hours spent at 1-sitting than Mathematics. Engineering trumps Computer Science as you can learn to be a programmer in your spare time.

*First know what you want to do as a career. Do not decide this on your own. Go to your career center for expert help, and shadow professionals and intern. Funny how this is #3.
Conclusion
Math can be a minor. Let your possible career path determine your major. At the least you can be earning a salary when you're 22-23 while deciding what new career you want. At the best, you'll save yourself 3-5 years of searching, finding-professional-self, and rebuilding qualifications.
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I decided to add some external links that would shed light on why some of my views are as they are. These come fro the Bureau of Labor Statistics... Oh look Statistics is even in the name =)
A small summary is included with the link, but please click and compare. 
These are just samples, you can find your own BLS data to refute if you want.
Occupations from mostly-pure discipline
1. Economist. $43/hr. 6% projected growth. 15.4K jobs. Bachelors Entry-level
http://www.bls.gov/ooh/Life-Physical-and-Social-Science/Economists.htm

2. Mathematician. $48/hr. 16% projected growth. 3.1K jobs. Masters Entry-level
http://www.bls.gov/ooh/math/mathematicians.htm

3. Statistician. $35/hr. 14% projected growth. 21.5K jobs. Masters Entry-level
http://www.bls.gov/ooh/Math/Statisticians.htm

Occupations heavy in STAT with Finance Augment
4. Financial Analyst. $35/hr. 26% projected growth. 236.0K jobs. Bachelors Entry-level
http://www.bls.gov/ooh/business-and-financial/financial-analysts.htm

5. Actuary. $42/hr. 27% projected growth. 21.7k jobs. Bachelors Entry-level
http://www.bls.gov/ooh/math/actuaries.htm


Conclusion for the sample
Stat has way more job positions filled/needed much faster growth than Econ, same entry-level as Math.
Finance augmented,  Statistically heavy jobs are quite abundant, pay well, and grow fast. This is what I recommended. I recommended majoring in Stat with a Finance augment. Both examples of this require only a bachelors and are in a job space projected to grow much more than the Mathematician or the Economist. The pure Statistician indeed makes less than those two. So to repeat from previously: Math Masters not needed if you like other things. Don't major in Math in undergrad.
