One filter I always have with me is a circular polarizer. It can give you more saturated colours, make an unattractive grey sky 'dramatic',
bring out clouds and take away reflections.
Nearly as important: a neutral density filter, which is not supposed to change the colours, just reduce the light. There are different types of such filters: either the whole filter is grey or it's graduated, which means it's much darker at the top and clear at the bottom.
There are many filter systems (like Cokin) which permit you to use a big rectangular filter and bring it exactly into the right position.
With a neutral density grey filter, this means that you can make the brightest part (in most cases the sky) darker, - which automatically means that the dark parts will be exposed longer.
A great way to deal with subjects that would otherwise be impossible to handle due to too much contrast.
Also, a neutral grey filter can be used to prolong exposure time, for example to give a special effect with
running water or to "empty" a public space of people. So, a neutral grey filter is always in my photo bag.
I sometimes used a filter to make a sky bluer than it actually was, but didn't like that one too much.
A "tobacco" filter can produce quite an interesting atmosphere though.
A very special area is the use of brightly coloured filters in black and white photography, but I'm not exactly an expert in that area (any volunteers?).
In any case: a filter is a bit of glass (or plastic) between your lens and the object.
This means that it will influence the "quality" of the lens. Cheap filters may also add reflections.
No use to buy that expensive high-tech lens and then use it in conjunction with a cheap filter - or even more than one!
To be on the safe site, I always shoot a subject with AND without filter and very often I prefer the unfiltered version.