Here's a picture of some clips:



They're made of a type of spring steel that when allowed to bend in its natural direction, the space between the teeth and the plate they're mounted on closes up. If you flex them in the opposite direction, they snap open a bit to make room between the teeth and the plate. So you'd pop them open before putting your system on your head, and imbed the teeth through your own hair a bit, then snap them shut by pressing on them through the system. The teeth on most of them are coated with a kind of silicon rubber to keep them from sliding off your hair once they're snapped shut. It's actually a pretty clever little mechanical device.

As John mentioned, if you wear a system all the time and use clips, you'll need to become adept at cutting them loose from time to time and re-sewing them back into slightly different positions to keep them from always tugging at your own hair in the same spots. From personal experience, you don't want to sleep in a system that is attached with clips as they will dig into your scalp and keep you awake. Even reclining in a chair or resting your head on the back of the seat in the car can be painful with clips. They can be visible through lace depending on where they are located or if they are shiny.