[possibly from gestures characteristic of stage magicians] To gloss over a complex point; to distract a listener; to support a (possibly actually valid) point with blatantly faulty logic.
If someone starts a sentence with "Clearly..." or "Obviously..." or "It is self-evident that...", it is a good bet he is about to handwave (alternatively, use of these constructions in a sarcastic tone before a paraphrase of someone else's argument suggests that it is a handwave). The theory behind this term is that if you wave your hands at the right moment, the listener may be sufficiently distracted to not notice that what you have said is wrong. Failing that, if a listener does object, you might try to dismiss the objection with a wave of your hand.
The use of this word is often accompanied by gestures: both hands up, palms forward, swinging the hands in a vertical plane pivoting at the elbows and/or shoulders (depending on the magnitude of the handwave); alternatively, holding the forearms in one position while rotating the hands at the wrist to make them flutter. In context, the gestures alone can suffice as a remark; if a speaker makes an outrageously unsupported assumption, you might simply wave your hands in this way, as an accusation, far more eloquent than words could express, that his logic is faulty.
A handwave conveyed the impression that the sea of troubles had risen to his chin.