[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.