There are tons and tons of filters, which all do different things. The main categories are:
UV filters - which don't do much on a digital camera. A lot of people use them for lens protection. There is a fair amount of debate on this, as there are a lot of people who swear by them, and then there are the people who say just use a lens hood unless you are going to the beach or a place with flying particulates. The cost of a front element and a really good UV filter are about the same. The difference in price often has to do with how optically good the filter is (how much UV it blocks as apposed to the rest of the spectrum, how much it is going to reflect light back into the camera reducing contrast, etc.)
Polarizing filters (and their cousins for use on AF cameras, circular polarizers) - they only allow light "vibrating" in one direction into the lens, there are a lot of great articles which describe this in depth, but the short of it is is that they are really handy for cutting down on certain specific reflections (like peering into water)
Color filters (which in the digital world have often been replaced with Photoshop) - These can be general things like warming or cooling filters, or correction type filters (correcting for tungsten, florescent, etc.)
"Shape" filters - these give a shape to highlights, like you saw in nocturns images of the child and the Christmas tree
Neutral density filters and graduated neutral density - these cut down on the amount of light coming into the camera. They are handy if you want to open the lens up, but you can't speed the shutter up to compensate. For instance, say you want a shot of a public square, but you don't want all the people. You could get a really heavy neutral density filter, and have a shutter speed of like an hour, which would blur all the people out. Graduated neutral density filters are the same, but they have a gradient from clear on one side to some amount of filter on the other side. These are handy for things like landscapes where the sky might be much brighter than the ground. Many people use HDR techniques instead of GND filters for more flexibility.
I'm sure there are others that I am forgetting that others can chime in on.