Why does a comet tail form when it nears the sun?

The exact composition of comets varies considerably, but the great majority of them come from the farthest reaches of the solar system and are basically just gigantic dirty snowballs, meaning that they are composed primarily of ice, but also have a few other things mixed in. They're too small to have an atmosphere or any significant gravity, so their cohesion and identity as a comet is dependent entirely upon them staying cold enough to remain in one piece. This was famously demonstrated by Shoemaker-Levy 9, which shattered into multiple pieces due to the influence of heat and gravity as it entered the inner solar system.

When comets form tails, they actually have two-- one of which is very visible, the other less so. The dust tail is less visible, and tends to more so in the direction that the comet is traveling. The ion tail always points directly toward the sun, no matter what direction the comet is traveling. Both tails can be thought of like steam coming off of the comet as it heats up, and the radiation from the sun interacts with it. Just as when you blow on hot soup and see the steam moving away from your breath, so too is the sun "blowing" on the comet, pushing hot particles away from it as it heats up, and those particles form the tail. These particles, which generally remain on the same orbital path as the comet, are typically responsible for seasonal meteor showers.