There are basically two kinds of astronomical filters: color filters primarily intended for use on planets (although they have some other uses), and used primarily to enhance contrast on certain kinds of nebulas. Nebula filters [2] can help in seeing through light pollution. Color filters do not.
But it isn't quite as simple as that....
Color Filters
Color filters are not helpful in overcoming light pollution. However, the planets are popular targets for washed-out astronomers and color filters are often helpful in pulling out certain planetary details (regardless of the light pollution conditions). So if you enjoy observing the planets, you may enjoy experimenting with color filters. Give them a try and see how they work for you.
Color filters can also be used with double stars to help bring out fainter secondaries if the two stars have significantly different spectral types (colors). In such cases, a filter that matches the color of the secondary may dim the primary and make the secondary easier to see. It doesn't always work, but when it does the effect can be quite spectacular. For example, try a blue filter on Izar (ε Bootis).
Nebula Filters
"Nebula filter" is a generic term for filters that attempt to improve the contrast of celestial objects by selectively passing certain frequencies of light and blocking others. They fall into two broad kinds. The first are “broadband” filters which pass most of the light frequencies and reject only a selected set of frequencies (usually including mercury and sodium vapor lamp light and neon light). Often called Light Pollution Reduction filters, examples of these the Orion SkyGlow and Lumicon Deep Sky filters. The reviews on these are mixed, suggesting that they have limited effectiveness in heavily light polluted skies. I was able to try one at a sky party under minimal light pollution and saw a distinct improvement in contrast on several nebulas and galaxies. But I’ve never tried one in moderate to heavy light pollution. Note that broadband filters are much more effective photographically than they are visually.
The other kind of nebula filter is termed a “narrowband” filter because they pass only a selected “narrow” set of light frequencies and block all other frequencies. The various narrowband filters differ in exactly the set of light frequencies they pass. Filters like the Lumicon UHC and the Orion UltraBlock are the “widest” of the narrowband filters, passing frequencies corresponding to doubly ionized oxygen and hydrogen-beta emissions. This makes them effective in enhancing contrast on a broad set of planetary and emission nebulas. Other narrowband filters are even more selective. O-III filters only pass the doubly ionized oxygen emissions, making them particularly effective on objects such as the Veil, Ring, Dumbbell and Orion nebula. H-beta filters are the narrowest of the narrowband filters, passing only the hydrogen-beta emission line. H-beta filters are supposed to be effective on the Horsehead, Cocoon and California Nebulae.
As a general rule, the narrower the filter the more aperture you need to use it effectively, the smaller the list of objects it is effective against, and the greater the potential contrast boost on those objects for which it is effective. It is also true (and ironic) that these filters work best under really dark skies. But they definitely improve the contrast even in heavily light polluted skies. My experience is that an UHC or UltraBlock filter works pretty well in enhancing contrast of most nebulas under washed-out skies. Note that narrowband filters don’t help at all with galaxies.