Simplify[expr] performs a sequence of algebraic transformations on expr, and returns the simplest form it finds.
Simplify[expr, assum] does simplification using assumptions.
Simplify tries expanding, factoring and doing other transformations on expressions, keeping track of the simplest form obtained.
The following options can be given:
Assumptions can consist of equations, inequalities, domain assertions such as x Integers, and logical combinations of these.
Example: Simplify[Sqrt[x^2], x Reals] .
Simplify can be used on equations, inequalities and domain assertions.
Example: Simplify[x^2 > 3, x > 2] .
Objects that are specified as satisfying inequalities are always assumed to be real.
Example: Simplify[x Reals, x > 0] .
FullSimplify does more extensive simplification than Simplify.
See The Mathematica Book: Section 1.4.4, Section 1.4.6 and Section 3.3.9.
See also: FullSimplify, Factor, Expand, TrigExpand, PowerExpand, ComplexExpand, Element, FunctionExpand.