Toe clips are a now-decrepit way of securing a cyclist's feet onto the pedals that said cyclist uses to propel his or her bicycle. These "toe clips" allow cyclists to pull up on the pedals, thus utilizing a different muscle group and improving efficency. Also, toe clips are good at keeping a mountain biker's feet from flying off of the pedals, and, by nature, keeping the rider attached to the bike, during insane rock and root-filled downhill singletrack trail sections.

Oh yeah, they look sort of like this:

 toe strap --> |     \ <-- plastic toe holder
                 ^-- axle of pedal

But, toe clips have since been made obsolete by clipless pedals. Clipless pedals are much easier to get out of (thus decreasing the potential pain of an endo) and look lots cooler. One caveat: The most popular style of clipless pedals for mountain bikes (the SPD), designed by the Microsoft of Bikes clogs up with mud very easily, and turns into expensive paperweights.

The only mud-proof options for ATB clipless pedals are the Time ATAC and Speedplay Frog models. I, personally, use the Time ATACs.