If you're playing one-player, all the other Links follow you around, and you can set their formations, and stuff like that. They're sort of like the "option" weapons from Gradius.
If you're playing four-player, then everyone gets to control just their own single Link.
If you're playing with 2 or 3 people, then you can actually fight with each other over who gets control of the extra Links.
Two-player means that each person logically gets one assistant, unless one of you wants to be a jerk and try to steal a second, leaving the other guy wanting.
Three-player is probably more fun, because there are three of you and only one extra Link, so there's always going to be someone with an advantage, and more people means that it's more likely that someone will eventually try to do something about it.