To squeak is to make a high, short sound. Your hungry pet mice squeak for food, and your creaky old wood floors squeak when you walk across them.
A door with rusty hinges will squeak when you open it, and your own voice may squeak when you're nervous. There are also the squeaks that small animals make, and the squeak that means "any communication," like when you don't hear a squeak from your next door neighbor for so long you wonder if he moved away. Informally, to "squeak by" or "squeak through" means to just barely achieve something, by a narrow margin.