Okay, so I know that using a color filter can have some desirable effects, for instance, if you use a certain color filter, it will diminish the effects of any skin problems a person has. What color filters (Digital or analog) can improve the overall appearance of a photo in what kinds of situations?
Don't use real colored filters on a normal digital camera - for a 15MP bayer sensor camera with a red filter, you would actually be shooting a 3.75MP image.
The exception is if you are doing IR or have a sensor modified to do only B&W photography.
In general for modern B&W digital photography your equipment is shooting in color, and then you decide after that what mix of color channels you want to use, so it's really up to you to decide the balance that looks best.
One common filter choice that people use is a red filter to darken the sky and enhance clouds. You can find pretty good descriptions of what filtering certain colors does for photography here:
Filters for B&W photography are really only applicable for negative film.
If you have a digital image you can achieve similar effects in post by manipulating channels during the B&W conversion.
Color filters help you to alter the way the various colors in a scene are translated to shades of gray.
In digital photography, don't put a color filter over your lens. You just decrease the light getting to the sensor, increasing noise for no good reason and decreasing the ability for the bayer interpolation to give you decent resolution. What's more, if you leave your camera on auto WB, your camera's white balance is likely to try and boost whatever colors you try and cut too reducing the effectiveness of what you were trying to achieve.
Instead, with digital you do this color filtering in post-processing. It has the same effect, but it's more flexible and cheap. For this you can use "channel mixer", or you can use "color balance" or "curves" or other tools on individual color channels to alter the relative color makeup before converting to black and white.
As others have said, there are certain limited situations where it may be desirable to use lens filters in digital photography, but it is usually for unrelated reasons than to affect the bias of colors for making a black and white image, which is what this topic is about.