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Different types of filters serve different purposes depending on whether you're talking about b&w photography or color photography. For b&w photography the typical filters are yellow, orange, red, green and maybe blue. Their purpose is to separate tones that otherwise would merge in a b&w photograph. In color photography different filters are used mainly to balance colors from different light sources and/or of a different color temperature or to create special effects such as a "warm" look.
Some filters are common to both types of photography. E.g. neutral density filters and polarizers are often used for both as are some special effects filters.
Many of the effects produced by filters, whether for color or b&w, can be replicated and improved in Photoshop. I used to carry a full complement of filters of different sizes for different lenses as well as adapter rings, brushes, cleaning fluids, etc. I no longer carry any of them except a polarizer and a couple graduated neutral density filters. I prefer to do what the other filters used to do in Photoshop where I have more flexibility than the traditional filters gave me. Obviously this is a personal preference, others may differ.
This is obviously just the briefest of summaries. To understand how different filters work and why particular ones might be used in certain situations you should delve into the subject more deeply than you'll likely get from answers here. There are plenty of books on photography, digital and traditional, that explain how and why filters are used and I'm sure there's also information available on line.
Having spent many years in a darkroom and now many years with Photoshop I'd have to disagree with mshi that you can do anything in a darkroom that can be done in Photoshop. That may be true in theory but in practice, no.