Why do monkeys have tails but people and apes don't?

This seems to be a common misconception - humans did not come from monkeys - but we do share a common ancestor with them. This common ancestor did not directly give rise to humans, it gave rise to the ape clad. These early apes divided into other evolutionary clad - we belong to just one of them.

The ape clad possesses certain shared traits that distinguish them from monkeys - one of these traits is not having a tail. The very first apes had no tails - a trait that is shared by all the species that descended from them.

Tails were lost in hominids prior to 17.9 Million years ago in response to the different ecological pressures experienced by apes. Ape ancestors did not hang from their tails (only South American monkeys can do this). In fact certain species of monkeys also lost their tails via evolutionary pressures.

Apes evolved powerful grasping hands and feet and moved through trees differently than tailed monkeys to take advantage of different resources. They eventually lost the extra bones beyond their coccyx that inhibited their locomotion.