What is a Planetary Nebula?

It's nothing to do with planets.

• "Nebula" is the Latin word for "cloud", and in astronomy any nebula is a cloud of stuff, usually gas and dust.
• Planetary nebulae* are a type of emission nebula.
• An 'emission' nebula is a cloud ('nebula') of gas that emits ('emission') light - i.e., it's a cloud that glows.
• 'Planetary nebula' is the name given to a shell of gas ejected by certain types of star (probably those between 0.8 and 8 Solar masses) in the later stages of their lives. The shell of gas is thrown off the star and expands. 

Why does this happen?

As you'll know from reading about the life of stars in other posts, when they run out of hydrogen to burn, the balance between the outward pressure of burning hydrogen and the inward pressure due to gravity is upset. The star starts to collapse in on itself, which provides the energy needed to start helium (which is there as a result of the hydrogen burning** phase) burning, and then the outward pressure is restored.

Actually, things aren't as simple as that. The helium burning phase starts before the hydrogen burning is completely finished, so you get a growing core of carbon and oxygen, surrounded by a shell of burning helium, which is in turn surrounded by a shell of burning hydrogen. The outer layers are cool in comparison to the inner layers, and they expand: the star becomes a red giant.

The helium reactions (going on between the inert core and the hydrogen burning outer layers) are incredibly sensitive to temperature changes which makes the star very unstable at this point, setting in motion a series of expansion and contraction phases which go a bit out of control: massive pulsations build up, which end up with the star throwing off its entire atmosphere into space. When the exposed surface of the star is hot enough (about 30,000 K), ultraviolet light ionizes the cloud and makes it glow.

And this is the planetary nebula.

Why the misleading name?

By ESO, via Wikimedia Commons

The first planetary nebula to be discovered was the Dumbell Nebula, by Charles Messier, and appears in his catalogue as M 27. He discovered it in 1764, and in 1781William Herschel discovered Uranus. Herschel noticed, through the low-resolution telescopes of the time, that M 27 and other similar objects looked a lot like Uranus. He then coined the term 'planetary nebula' for them, and the name stuck even though we now know that they are very different objects to planets.

What's so important about them?

It's probable that we owe our very existence to planetary nebulae. They serve a very useful function: just as all the elements heavier than helium have been manufactured inside stars, so planetary nebulae provide a route by which they return to the universe at large to be included in the formation of heavier-element stars and their accompanying solar systems. The heavier elements, known as metals by astronomers, are essential for the eventual formation of life like us: without them we wouldn't be here.

*'Nebulae' is the word for more than one 'nebula'. Think 'nebulas' only posher.
** 'Burning' isn't the best word to use here, really. Stars don't 'burn' like a fire does; they produce energy by way of a process known as fusion in which atoms are 'fused' together and form different ones: if you smash hydrogen atoms together in the right conditions they 'fuse' to become helium atoms. When the helium 'burning' phase starts, the helium atoms fuse to form a mixture of oxygen and carbon.