What makes the earth spin?

Short, simple answer is that when planets form from gases, there is usually some rotation of the gases, and as they collapse the spinning is accelerated... much like a skater spinning with her arms outstretched and then brought in... as the planet forms, the original rotation of the gases is maintained.

To amplify on kp2171's answer - the property of "conservation of momentum" causes the earth to keep spinning. The earth spins because the gas & dust cloud it coalesced from 5 billion years ago had angular momentum. In fact this original angular momentum is also the reason why all the planets orbit in the same direction, and with the exception of Neptune all planets and the sun have their axis of rotation roughly aligned and rotate in that same direction. Since momentum is conserved, the earth keeps on spinning, and will forever unless there is some force applied to the earth that opposes the rotation. Actually, tidal forces actiing on the earth due to the moon, and to a lesser extent the sun, do in fact oppose the earth's rotation and hence the earth is slowing down. That's why every few years they throw in a "leap second," to make up for the gradual slowing of the earth's rotation.

In fact, we have a leap second coming up. On December 31st this year, there are 2 seconds between 23:59:59 and 00:00:00, instead of the usual 1. People counting down to new years should say 1 twice ;).