What is this game with 16 pegs and 36 spools to fit on them?
This looks a version of a game I used to own called Score Four. It is essentially the classic Connect 4 game, but in three dimensions instead of two. 

It plays exactly as you would imagine


players take turns adding pieces of their color onto pegs
the first player to create a line of 4 in any direction (vertical, horizontal, diagonal) wins


It is considerably more interesting than Connect 4

http://boardgamegeek.com/boardgame/3656/score-four