Most filters are obsolete in the digital era, IMO. The effects can be very easily replicated in post production, without the limitations of physical filters.
That being said, you can't replicated all the things that a polarizing filter can do, in post production. So a polarizer is still a very useful filter to have.
Another group of filters that can still be useful, are graduated (grad) filters. They are dark on one side and clear on the other. There are various types; soft edge, hard edge, more or less darkening etc. There are color versions as well, but color effects can be done in post. They also come in round or square shapes. The round one don't interest me at all, as they will force your composition. The square ones are more adjustable, and thus more flexible.
So, you may be thinking that you can easily darken part of your photo in post, so why do we use these filters? They help to compress the dynamic range of the scene (as seen by the camera) so that the camera can capture detail in parts of the scene that may otherwise be too dark or too bright to capture detail....do to the limited dynamic range of our cameras.
Another useful filter is the Neutral Density (ND) filter. It just darkens the whole scene, which allows you to adjust your exposure settings. Usually to get longer shutter speeds.
And yet, some people use UV filters. The UV part isn't necessary, as digital cameras aren't really sensitive to UV light...but because the filters are basically clear, they are used for physical protection. Better to damage the filter than the lens. I personally don't use them, but many do.
A key point when talking about using filters or not, is how much they will affect your image quality. Technically, anything you put in front of your lens will potentially affect your image quality. You want to avoid the cheap filters because they will certainly hurt your image quality, and increase the chances of thing like lens flare. Better filters have coatings, some of them are MC (multi coated) to help reduce flare etc.