Difference between Pseudorandom Function vs randomly chosen function
Random function -- function $F$, that is selected randomly from the set $Func$ of all possible functions (with given domain and range).

Pseudo-random function --- family $\{F_k\}$ of functions, that is indexed by the parameter $k$ (which serves as a number). It is pseudo-random, because if someone picks $k$ secretly and lets you interact with $F_k$, it should look like you are working with a random function, whereas in fact it is chosen from a much smaller set, not from the set of all possible functions.