Pseudorandom functions: how are functions stored?
For the definition of pseudorandomness, the family $F$ of functions can be any set of functions at all. But typically we take it to be a set where each function can be described by a rather short key/seed, and where one can efficiently compute the function output given the input (and the key). This is because we want the family $F$ to represent functions that we can randomly choose from and use in real life.

For example, $F$ could be the set of functions AES$_k$, taken over all 128-bit strings $k$ (where AES$_k$ denotes the AES block cipher with key $k$).  Notice that there are "only" $2^{128}$ functions in this family, which is much less than the number of functions mapping 128 bits to 128 bits (which is $(2^{128})^{2^{128}}$).