To complete is to make a bet in order to match the forced bet balance on a table, or to match all-in raise balance (where the raiser does not have the required chips for a minimum raise).
Player A raises by 40 chips.
Player B would have to wager 80 chips for a normal minimum raise: the 40 to match player A plus an additional 40, but he only has 60 and goes all-in with these. A can only complete by matching B's 20 chips. If a third player were in the hand, A could, depending on the house rules, decide not to raise again becasue B's raise was irregular; A can only complete it.
In games with blinds, a complete is when the small blind pays the difference between the small blind and the big blind to stay in the hand, provided there have been no raises by the time it is his turn. The big blind is a complete bet, i.e. the amount that a player must wager to stay in the hand. Since the small blind only pays half the big blind, his bet is incomplete.
In games with an ante structure, a player can choose to bet the full amount instead of the smaller bring-in.