What are lens filters?
They are squares or rings of optical glass which you place in front of your lens to modify the light entering the camera.
How do they attach?
Circular filters attach to the front threads of your lens, to see what size your lens's threads are, look at the front of the lens, there is usually a marking. Square filters are used in a matte box, which is a larger more involved piece of equipment.
You get what you pay for still rules much of the time, but there are some very good bargains out there. Do NOT just go to a camera store and pick up brand X, look for well reviewed filters. If it's really cheap, it's probably no good. Think of how silly it would be to put a poor piece of glass in front of your expensive lens?
What types are there?
Hoya's are a good balance of price/performance: Hoya HMC UV Multi-Coated Slim Frame
The most basic type which is more or less considered protection for your lens, because replacing a filter is much less expensive than replacing/fixing a lens.
I mostly use the 3 stop Tiffen ND 0.9 Filter
Neutral Density filters just cut the amount of light going into the camera. FIlmmakers many times want to use a wide open aperture to get a shallow depth of field, and in a bright situation an ND filter can be the only way to accomplish this properly. They come in different strengths measure in stops, just like F stops. So if you happen to be in a bright sunny day and are going with the sunny 16 rule, you'd need a 5 stop ND to get from f/16 to f2.8
I've got this one Fotodiox Graduated ND
Ever wonder how to get a bright sky and a dark landscape properly exposed? One way is to use a grad ND, which is ND on one half, and clear on the other. The middle point fades between the two, so you can dim one half of the image and maintain the other half. Works best with non-moving shots. There are also colored version should you want to stylize/modify further
I use the Marumi DHG Super Circular Polarizer
This is one of those magical things, and you shouldn't leave home without it. A polarizer cuts out various angles of light. This means it can reduce or eliminate glare. Reflection in the water? Polarizer. Reflection in a window? Polarizer. It won't work 100% of the time because it's dependent on the angles of reflection, but many times it can be a dramatic improvement. It also can saturate colors, and deepen blue skies. They have a rotating front element so you can adjust how much of the effect you want.
This one is relatively inexpensive and worth a look Light Craft Workshop Fader ND MK II
These are basically two polarizers paired to create a ND filter which can be dialed up and down in strength. They sound ground, but there's no free lunch, many have color shifts and other undesirable effects. There are some very good ones out there, but most are plenty pricey
This is a pretty complete set Fotodiox 7 Metal Step Up Ring Set
Now, a simple trick to not have to buy a filter for every size lens you have, just grab some step up rings so you can buy 1 larger filter, and the rings allow you to attach them to smaller lenses. You can buy a whole set, or just find what you need. They're plenty cheap no matter how you slice it.