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Ultra violet filters
These filters are a must have. Not because they do any fantastic thing to your photo or anything, but because of what they do for your lens. An ultra violet lens is essentially a piece of glass, nothing else, just a clear piece of glass. This means it will not change how your picture looks the least. What it will do is to protect your lens from dust and scratches. If you accidentally clean your lens a bit hard and scratch it, with an ultraviolet filter on, you take it off and buy a new one, loosing about $20, without it you just ruined your $500 lens. Lenses can cost you quite a bit more than $500 so investing in an ultra violet filter is a cheap insurance for your lens. I have bought one for each lens I have and they stay on permanently. As you can attach other lenses while keeping the UV lens on there is no reason to take it off. So I strongly recommend getting them, one for each lens.
Circular Polariser
This is an interesting filter. 
Essentially it is sunglasses, but only half of what sunglasses does. Good sunglasses polarize the light, meaning they cut the 3 dimensional electromagnetic wave in half. (Thus if you put two polarizing lenses at a 90 degree angle no light gets through). While sunglasses often have a colour as well, the CP does not. Now why would you want to have this on your camera? To create really vivid blue skies! If your photo has some sky in it, I would consider putting on your CP, it also affects other colours making them look nicer. Another thing that is nice with CP is that it can remove reflections from glass. This can be really handy if you take photos through a window. Note though that it may not always work as it is depending on the direction of lights, the curvature of the window and lots of other factors. The CP lets you adjust how much polarizing you want by rotating the front of the filter. As the filter reduces the amount of light that comes in through the lens it will affect your shutter speeds. This filter is not essential but I recommend getting one as they are often quite useful.
Neutral Density
This filter is a filter that when you first hear about it you think 'Why would I want that?'. The filter comes in various grades and it is just flat grey. It will not change the colour composition of your image, all it does is to reduce the light coming in. I have the darkest neutral density I could get, it will reduce shutter speeds significantly giving me second long exposures in daylight. This has a very specific application for me, moving water. With this you can get great shots of moving water. There are other applications as well, I am sure, so if you ever find you would like to extend the shutter time but not getting washed out colours, neutral density is your answer.
Other filters
There are plenty of other filters. Some giving you strange effects, others changing the colours of the photo. Infra-red filters lets you take photos in infra-red. Sepia filters will give you sepia colours in your photo. There are plenty other filters so if you find you would like to do something have a look if there is a filter that can do it for you. Filters are in general cheap so you can get one and try.
Multiple filters
As I mentioned above filters can be attached one after another. This is very handy, when I was up the mountains on a bush walk I had my circular polariser as well as my neutral density on the lens that gave me an 8 second(!) exposure outside on a day with almost no clouds. As I said before as well I always keep my UV filters on to protect my lens. So just because you want to use one filter does not mean you can not use a different one as well.
Filter vs Post processing
Some people will claim that most of the things I can do with a filter can also be done after the photo is taken using programs like Photoshop, the gimp or similar. True getting a sepia filter may seem a bit silly as you can take the photo in full colour and then change it if you realize you want it in sepia. The thing though is that with a filter you get a much better photo compared to doing it afterwards. Having said that, the post processing does not give you a bad photo, but a filter will give you a better photo. Simply because everything in your camera will be adjusted to the changed conditions. Your light meter will measure the "sepia" light maybe indicating that you should adjust the shutter speed a bit. In general whatever you can do in optics will give better results than doing it in software afterwards.
