The term quantum refers to discrete units of energy and matter at the subatomic level. Quantum mechanics is the study of how matter and energy interact on very small scales. There are many interesting quantum effects that can occur, such as particles being in multiple places at the same time, traveling through time, moving through solid matter, being created from nothing, disappearing into nothing, etc. These effects cannot normally happen on macroscopic scales (larger than an atom or so). Technically, they could happen on larger scales, but the probability is so low that it would realistically never happen throughout the history of the universe. This is one reason why characters that can manipulate probability are so dangerous.
Quantum is also a popular technobabble word, often inserted into random jargon to make it sound "scientific". Star Trek, particularly Voyager, is notable for doing this.