C[i] is the default form for the i constant of integration produced in solving a differential equation with DSolve.
The indices i are chosen to be successive integers, starting at 1 every time DSolve is called.
The function applied to the index i to produce the i constant of integration is determined by the setting for the option DSolveConstants in DSolve. The default is DSolveConstants->C.
See The Mathematica Book: Section 1.5.8 and Section 3.5.10.
See also: Unique.