I'm not an expert on filters, this is a learning experience for me. As such, I may have some of this wrong because I haven't read some books on the matter to have a clear understanding of all the effects and interactions. Take this info with a grain of salt; however it is what I understand that things work.
I suppose that you could argue that all filters are color correcting! That is true to a certain extent; however the Color Filters adjust the color temperature of light to put it where it is expected. The Color Correcting or CC filters are used to correct actual color casts which are present, not to affect the daylight/tungsten.
One use of CC filters is to compensate for the green tint which fluorescent lights add. You can use a Magenta CC filter (RGB CMY) to subtract the extra green light added by fluorescent lights. There are two special purpose filters which do this. One is the FLD filter for daylight fluorescent tubes. The other is FLW for warm flourescent lighting. There may be other filters available for the other casts of fluorescent lighting as well.
One problem though. If you filter your lens to get rid of the extra green, then any flash use will cause a different color cast! To take care of this problem you need to filter your flash to be the same color as the fluorescent lighting. Fortunately you don't need optically perfect filtration here! One source suggests a Green CC037 filter as a possible color match for fluorescent lighting when using a strobe.
The B&W filters also have names such as O1 O2, Y1, Y2, R1, R2, and G1, G2. The letter is Orange, Yellow, Red, Green, and the number is the density of the filter. I don't have the equivalents of the numbers below to those names, but it seems quite useful to find.
For B&W filter use the TTL meter in the camera and other hand-held meters (unless equiped with filter compensation) may not correctly expose the film with a filter. This is because the meter has a different sensitivity to some light wavelengths than the film does. I have heard that this is because of the red light affecting the film more -- the redder the light, the more light is required for a proper exposure. This is NOT a problem with color film as it has a sensitivity similar to the meter.
Thus, it is important to know what the filter factors are for B&W filters: I don't know if these factors are for all filters of a color, or just for the darker '2' variants:
One procedure for working with filters is to first examine how many stops they effect the in-camera meter by. Say, for example, that the orange filter affects the camera meter by 1 stops. That is how much the density of the filter effects the meter. However, there is still the total effect the filter will have with the film. Using the one stop density adjustment as a factor, you would then dial in the additiona 1-1/3 stops of EI correction by using exposure compensation or by adjusting the film speed by the same number of stops. The end result is that when your meter is at 0, the exposure will be correct for the filter you are using -- just as if you were using color film. Remember to adjust these offsets when you change filters!
Some meters, however, may expose correctly for filters -- presumably they have the same response curve as the film. Presumably this is true only for much older cameras, or for certain types of meter sensors. Probably the best thing to do is to test your equipment and see how it performs with each type of filter.
After reading some more it appears that Panchromatic B&W films respond differently to filters than do Orthchromatic B&W films. In other words, the filter factor varies depending upon what the film type is. The Kodak Professional Photo Guide is recommened as a reference for details of this nature.
I'll need to rewrite this section soon. It turns out that Chromogenic B&W films --- aka C-41 process B&W films have a different color response than the traditional pan or ortho films. Panchromatic film is blue-sensitive, hence the need for longer exposure times when reddish light is used. Chromogenic films are more Panchromatic than the Panchromatic films, and thus the filters can affect them more than they affect silver-based B&W films.