Hesitate, waver, vacillate, falter all mean to show irresolution or uncertainty.
Hesitate, the general term, usually implies a pause or other sign of indecision before one makes up one's mind what to do, say, or choose.
Waver (see also SWING 2 ) implies hesitation after a decision has been reached and so usually connotes weakness or retreat.
Vacillate implies prolonged hesitation resulting from one's inability to reach a fixed or final decision; the term connotes alternate decision and indecision or a shifting (as in opinions, choices, or loyalties).
Falter suggests a wavering in purpose or action that is evident or is made evident in such signs of fear or nervousness as trembling or the breaking of the voice.