The term "clip in" refers to the act of engaging a metal cleat, attached to the bottom of bicycling shoe, to a specially designed pedal. This firmly connects the shoe with the pedal, reducing or eliminating slippage of the foot while pedaling, and allowing the rider to pull upwards on the pedal as it comes around to complete a revolution. This in turn allows for more power to be gotten out of each pedal stroke, which is especially handy when riding aggressively (as in bike racing, or when charging uphill, usually "out of the saddle").
To "unclip" from the pedal, the usual method is to twist the ankle with an outward motion, which causes the pedal to release the cleat. Making this motion instinctive is essential to avoiding injury when there is a sudden need to disengage from the pedals, such as coming to an abrupt stop or when a Very Bad Thing occurs with the bicycle itself. As an example, I was commuting one day with a backpack. One of the backpack strap buckles snapped, and the pack swung around on the other strap, slamming into my bicycle's front wheel in an instant. Before the backpack had even swung halfway around, I had instinctively unclipped from both pedals and brought the bike to a safe stop. If not for that, I probably would have taken a nasty spill in the middle of the road.
Examples of pedal systems you clip into include Look pedals and Eggbeater pedals.