What math branches are there? This might be too general question, but still, I think this may be some useful question.
So, what math branches are there generally? (for example, one branch would be abstract algebra, differential geometry etc.)
Or what math branches are recognized in undergraduate math programmes?
 A: One way is to look at how mathematicians divide mathematics. The 2010 MSC is the latest version of a series of attempts to categorize all of mathematics; its first level should give some kind of an idea of what the branches are.
http://msc2010.org/mscwiki/index.php?title=MSC2010
A: This is just my opinion and how I think about it.
Your question is not a bad question, but it is difficult to answer because it is going to depend on peoples personal feelings. One might "divide" math into areas in one way, while another might divide another way. Besides that, there is the question about how deep you want to make your division - i.e. how many areas are we talking about. Then there is the "problem" that with pretty much any division there are going to be overlap. So if you go to conferences, you might see a person going to different conferences in different areas because he/she works in an area that that overlaps two other areas. So it is really diffictult.
However, when I try to explain what I do, I usually start out by diving math into the areas


*

*Algebra

*Analysis

*Geometry/Topology

*Applied math


(Yes, some people like to put the applied math under analysis).
Now you might ask: "So what do these 'different' areas each cover?" or "What distinguishes one area from another?". This is even harder to give a good non-offensive answers to, so I wouldn't even try.  
A: Here is one of the most amazing things ever written: http://en.wikipedia.org/wiki/Lists_of_mathematics_topics
This differs from other lists that might answer such a question as this one, as a result of the differences in origins and purposes.
Consider that the following is just one small section, and look at what's in it:


*

*List of convexity topics

*List of dualities

*List of exceptional set concepts

*List of exponential topics

*List of factorial and binomial topics

*List of fractal topics

*List of logarithm topics

*List of numeral system topics

*List of order topics

*List of partition topics

*List of polynomial topics

*List of properties of sets of reals

*List of transforms

*List of permutation topics


Another small section is "Work of particular mathematicians":


*

*List of topics named after Augustin-Louis Cauchy

*List of things named after Albert Einstein

*List of topics named after Euclid

*List of topics named after Leonhard Euler

*List of things named after Paul Erdős

*List of topics named after Fibonacci

*List of topics named after Carl Friedrich Gauss

*List of things named after Charles Hermite

*List of topics named after Joseph Louis Lagrange

*List of topics named after Srinivasa Ramanujan

*List of topics named after Bernhard Riemann

*List of topics named after James Joseph Sylvester

*List of topics named after Alfred Tarski

*List of topics named after Karl Weierstrass

*List of topics named after Hermann Weyl


Here's the section on geometry and topology:


*

*List of geometry topics

*List of geometric shapes

*List of curve topics

*List of triangle topics

*List of circle topics

*List of topics related to pi

*List of general topology topics

*List of differential geometry topics

*List of algebraic geometry topics

*List of algebraic surfaces

*List of algebraic topology topics

*List of cohomology theories

*List of geometric topology topics

*List of knot theory topics

*List of Lie group topics

*Glossary of differential geometry and topology

*Glossary of general topology

*List of points

*Glossary of Riemannian and metric geometry

*Glossary of scheme theory


Click on "List of circle topics" and marvel at it.  (Full disclosure: I'm the principal author of that one.)
These are only a few of the sections.
