I'm sure there are more answers if you search the forum, but the search is not always helpful. Anyway:
bright - (over)emphasis on high frequencies. This gives the impression of greater detail and clarity, but may be tiring for long listening sessions. These types of speakers usually need a "warm" amp to compensate, but a better (and more expensive) solution is to find an amp that is maddeningly accurate in the higher frequencies. The opposite is dark (not warm)
warm - (over)emphasis on upper bass and mid frequencies. This tends to improve voices but usually at the expense of clarity. The opposite is lean
forward - akin to bright, but said more of amps. In my mind, forward has more to do with energy delivery (how "lively" an amp sounds) than with actual frequencies. The opposite is laid-back
I'm sure others can explain better than I do, and I hope I've not made too many errors that others will need to correct.