My opinion (UK ACC)
Expedite to me means max. continuous power and best angle of climb. I use it once or twice a shift. Like Scott said (paraphrasing), I use it when I think I can avoid using headings e.g. catch-ups. Normally I qualify it e.g. "Descend FL100, Expedite through FL160". Effectively it means do better than you are already!
If I've screwed up, I'll say "Climb/descend IMMEADIATELY" which means TOGA power, control column into beer gut/firewall and pray!
Pilots - we are aware about Mmo, how you slow up in the descent, and your dislike of speedbrakes and we take this into account. With climbs we know (a bit) about flap limiting speeds and your profile, because we see you on the radar every day.
Oh, and I never ask an A342 to expedite anything. 800fpm at FL100, they just get the vectors. Sometimes its useful to let the things level off at lower levels, at least that way they start accelerating.