A DNF (short for Did Not Finish) is the result of a solve that was not completed successfully. The typical usage of this term is for any solve where the timer stops when the cube is unsolved, which can happen if a puzzle has a pop and the cuber does not want to reassemble the cube because the time will be too slow, or in a blindfold event where the puzzle is not correctly solved. There are also several other situations where a solve can be given a DNF, which are all outlined in the WCA regulations. Note that a DNF is considered to be a worse result than any time or amount of moves, so in an average of 5 it will be removed along with the best solve, but if there are two or more averages than the average becomes a DNF as well. In a mean of 3, even one DNF will make the mean a DNF. A similar concept is the DNS.
In BLD events, the DNF plays a very large role, because even the best blindfold solvers will sometimes make a mistake and end up with an unsolved cube at the end. Since DNFs are so common - some solvers may take five or more competitions to get a successful solve - blindfold events in competition never have averages, but are always in the 'best of' format. The possibility of DNFs can make it very difficult to set blindfold records: on one hand, to set a record it is usually necessary to go as fast as possible; but on the other hand, going very fast decreases accuracy, so it is possible for someone who is going as fast as possible to get no successes in the entire round, and thus end the result with a best result of DNF.
DNF is also the default result for a time that has been invalidated. For example, Grzegorz Prusak set a Square-1 single record once which was so easy (the scramble was incorrectly done) that he later agreed to have it removed. Thus that solve is now a DNF on his official profile.