Simplification of multiple cases (the quadratic equation had 5 forms until negative numbers were introduces)
Removing technicalities (the Lebesgue integral fixed the fact that differentiation and integration were not truly reverse procedures)
Necessity out of prior concepts (Solving the cubic formula forced the existence of Complex Numbers)
Abstraction from material ideas (numbers, geometry)
Uncovering the true underlying assumptions (non-Euclidean geometry was created not as a new geometry, but rather as an attempt to prove the parallel postulate)
Approximations
Abstraction from ideas occuring many times in Mathematics with an attempt to find a complete theory to describe them (What are groups?, Topological Spaces [to write])