In the context of matters Christian, the term charismatic refers generally to the exercise of the gifts of the Spirit. This generally includes such things as:
Speaking in tongues
Sometimes also known as "ecstatic babbling," or "glossalalia," this does not normally involve the use of known languages. When treated as a "language," it is usually referred to as a "heavenly" or "angelic" language.
Translation of "tongues"
Most often referring to someone exhorting a congregation with a "word from the Lord" rather than a prediction of future events. (Note that this is consistent with the Jewish Old Testament; most of the "prophetic" books were primarily directed towards telling Israel to clean up their act.)
Being an apostle
A charismatic is someone who holds to the doctrine that the above list of gifts do apply to Christians today. This view is particularly characteristic of Pentecostals. Some take the more extreme view that an individual must speak in tongues in order to evidence salvation, which I believe is contrary to what the Bible says.
At the other end of the spectrum lie such groups as Baptists, who hold to the doctrine that these gifts were intended for a particular time early in the history of the Church, but that that time has passed, and that the charismatics are just fooling themselves when they practice so-called "gifts."
To be flippant, I could comment that Baptists tend to be rational, overly-self-controlled, and boring, whilst Pentecostals "get insane" and (at least in worship) have greatly more fun. And Mennonites may talk a lot about peace, but their "true colours" show when they play the card game Dutch Blitz. That of course would be an unfair overgeneralization of all groups, which can be argued to average out to being fair.
I don't find the reasoning of either the strongly charismatic or the strongly anti-charismatic to be completely compelling, and so cannot quite comfortably sit 100% in any of these camps. (And am "bloodthirsty" enough to establish that I am not a Mennonite.)
Note that I grew up in an "independent" church in Ottawa, Canada that essentially holds to the non-charismatic stance, spent 1993-1996 at a (not highly charismatic) Pentecostal church in Toronto, and most recently am at a Baptist church in Texas.