size_t is the type used to represent sizes (as its names implies). Its platform (and even potentially implementation) dependent, and should be used only for this purpose. Obviously, representing a size, size_t is unsigned. Many stdlib functions, including malloc, sizeof and various string operation functions use size_t as a datatype.
An int is signed by default, and even though its size is also platform dependant, it will be a fixed 32bits on most modern machine (and though size_t is 64 bits on 64-bits architecture, int remain 32bits long on those architectures).
To summarize : use size_t to represent the size of an object and int (or long) in other cases.