As someone else has already detailed in the aji node, aji is a Japanese term from the game of Go (Wei Qi/Badouk). It means "taste" but a better English word would be something like potential. If a certain group of stones has aji, it means that there is potential for an attack there at some stage of the game.

Aji-keshi basically means "destroy own potential." A move which is aji-keshi is bad, because the opponent, in responding to it, can simultaneously fix up an earlier weakness in his position. A good example of aji-keshi is a peep at a cutting point. Consider the following position (if you don't know the rules of Go, check that node out first so that you can understand the diagram):


Black's (x) position has a potentially devastating weakness at the spot marked a. If White (o) plays there, the Black stones on either side are both in atari, so one of them will be captured. Imagine, though, that white plays one to the left (b) of that point instead. He is peeping at a cutting point, and this allows Black to play at a himself and fix the weakness. This is obviously aji-keshi.

Of course, only a total beginner would even consider playing at b (unless there was something about the surroundings that made it favorable), but I wanted to use something obvious as an example. It is important to realize, however, that sometimes aji-keshi is not so obvious, and sometimes what looks like a good move at first actually turns out to be aji-keshi.