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A quick note: just because your camera doesn't have a threaded filter mount doesn't mean you can't use filters. There are a number of ways to DIY filter mounts, be they for purchased filters or those you make yourself. So, don't fret if you have a point and shoot but reeeeally want to get into the whole filter thing: it's possible, you just need to dedicate a little time into crafting the right tools.
That being said, my top three filters!
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Now, UV penetration may not be the biggest concern of every photographer. After all, many modern lenses already come with a UV protective coating. As far as aesthetic changes to the photo is concerned, a UV filter will reduce something of that hazy glare from the Sun.
However, the major use for a UV filter is not so much the UV aspect of it, but the fact you have a cheap, easily replaceable filter between your much more expensive and difficult to replace lens (and expensive filters) and the elements—be they approaching Saharan sandstorms at sunset or just a little dust blown up by the back of a pick up truck or a few spots of rain. It takes out some of the worry when shooting in relatively dangerous conditions for your camera. That's not to say you shouldn't take all the important precautions, but it should free up the photographs you can take. $5 is worth this, yes?
And if it breaks, don't fret about that either. Scratched and damaged UV filters are easily convertible into filter mounts for any number of DIY filters.
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You need a polarized filter. You need a polarized filter. You need a polarized filter. Should I say it again? You need a polarized filter.
This is the single most versatile lens in your arsenal. The primary use of a polarized (or polarizing) filter is for minimizing light coming in from a single direction—especially handy for those overblown sunny skies and hyper-reflective skyscrapers. With skies, it also works to deepen and intensify the colour, from a robin's egg blue to something closer to an azure, and bring out the contrast within cloud formations. For vegetation, the colour is enriched, making it seem more vibrant and alive. It can even cut down on those sometimes pesky reflections in windows and other reflective surfaces, providing a crisp view of the other side.
This is also one of the few filters that simply can't be simulated using digital editing technology—the direction from which the light enters the digital sensor simply isn't encoded in any image file format.
DIY polarized filters are also a possibility, either for budget concerns or for a general love of building things.
If you can afford two polarizing filter, you can even use the polarized filters to double as another important filter: an adjustable neutral density filter! By spinning one relative to another, you can go from blocking one direction of light to all directions of light as they go from parallel to perpendicular. Your photo, thus, is darkened or as lightened as much as you want, and you even have control over which light is getting effected. This is an especially potent tool for working all the usual neutral density filter places, from long exposure waterscapes to intensely bright sunny days. The details of this are described in the second half of the DIY neutral density filter article.
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Graduated Neutral Density Filter
Your last must-have filter is the graduated neutral density filter. This isn't as versatile as the other two filters, and really, you only need it if you're planning on getting out and about to do some serious work with both sky and land in the same shot—a given for most photographers in the great outdoors. Think sunsets over the sea, dawn over the mountains, the like.
For such shots, it's incredibly difficult to get both the heavens and Earth exposed correctly. In fact, it's often impossible to do this on the same shot... unless you have a graduated neutral density filter handy.
Basically, graduated neutral density filters are a piece of filter that go from dark to light with some sort of gradient. This gradient can be either abrupt, or over a wide space. They may also go from clear to varying levels of darkness, from near pitch black to a subtle pale grey. Between these two variables, you've got a lot to choose from.
Now, unless you want to get a whole set (expensive), you'll be somewhat limited with the one you'd be purchasing. I'd play around and see what sort of composition you prefer for such shots, and thus what sort of graduated ND filter would be best for you.
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Now, that's not to say all those other filters aren't cool. They're just highly specialized. Infrared filters, star filters, UV pass filters, diffusing filters, all those sorts usually don't have any daily use for a photographer. Focus on getting these three critical filters first before moving on to these—the basic effects before the exotic!