Apparently this is a simple question with a not-so-simple answer.
I believe the general consensus is that there is a thin layer of liquid water on the surface of the ice. This thin layer and the solid ice below it are responsible for the slipperiness of ice; the water easily moves on the ice. (Well, why is that? Perhaps another SE question.)
However, this is no real agreement as to why there is a thin layer of liquid water on the surface of ice to begin with. See here for a 2006 NYT article. And if are interested in the actual physics paper that the news article is based on, see here (DOI).
One idea states that the molecules on the surface of ice vibrate more than the inner molecules, and that this is an intrinsic property of water ice. Since the outer molecules are vibrating faster, they're more likely to be in a liquid state.
Another idea is that the movement of an object over ice causes heating, though I found conflicting sources as to whether there's a consensus on this.
There is a popular idea that many hold but doesn't appear to hold water. (Heh.) This idea posited that the added pressure on the ice from a foot or skate causes the melting point to rise, which would cause the thin layer of liquid to form. However, calculating the resulting pressure and increase in melting point doesn't line up with observation; the melting point certainly does rise, but not enough.