The axiom of choice says that if you have a set of objects and you separate the set into smaller sets, each containing at least one object, it is possible to take one object out of each of these smaller sets and make a new set. You do not always need to use the axiom of choice to do this. You do not need to use the axiom of choice if the starting set is finite, or if the starting set is infinite and has a rule built in for how it can be divided. For example, for any (infinite or finite) sets of pairs of shoes, one can pick out the left shoe from each pair, but for an infinite collection of pairs of socks, the axiom of choice is needed.