There are many "types of mathematics":
algebra - dealing with calculating numbers and graphing functions
geometry - shapes, solid figures and geometric plane
trigonometry - studying triangles and relationships between their sides; also includes trigonometric functions, such as sine and cosine.
calculus - limits, derivatives, integrals, infinite series and functions
linear algebra - vectors
combinatorics - study of finite and countable discrete structures
differential equations - functions with one or more variables and derivative orders
real analysis - set of real numbers
complex analysis - functions of complex numbers
abstract algebra (includes group theory, ring theory, field theory, and module theory) - dealing with algebraic structures, such as fields and rings
topology - relates to geometry and set theory; also deals with shapes
number theory - numbers (that's obvious!)
logic - related computer science and philosophical logic
probability - occurrence of events (for example, rolling dices)
statistics - interpreting data
game theory - applied mathematics; "games" interaction (I'm not very good in explaining this)
functional analysis - dealing with abstract methods to study algebraic problems
algebraic geometry - combination of abstract algebra
differential geometry - part of integral and differential calculus
dynamical systems (includes "chaos theory") - another applied math; connections to physics, economics, biology and philosophy
numerical analysis - dealing with algorithms
set theory - dealing with sets
category theory - connecting to abstract objects and arrows
model theory - study of mathematical structures, such as fields and groups
I hope this helps!