Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
For an overview, see the Wikipedia entry on Discrete mathematics.
In my eyes defining discrete mathematics is not far, difficulty wise, from defining mathematics. This is partially due to the broadness of it. This very broadness is one of the reasons why I think this tag is bad. (We don't have a (mathematics) tag, do we?).
Often times questions are tagged (discrete-mathematics) alone which is not saying much and when they are tagged in conjunction with other tags, then discrete mathematics tag becomes superfluous. This is so common that I think it's not necessary to provide examples.
Another problem is that this tag is very attractive due to the fact that its name is the name of the very courses people take. This alone wouldn't be a problem, there are other tags whose names are common courses' names too. The difference between (discrete-mathematics) and those other tags is that the other tags' names are specific enough to be used as tags.
I believe it should be deprecated.