Generally, staking means that one or more persons put up money for a poker player to play with. In the best case, the staker receives the sum of his "investment" back and in addition to that, a predetermined percentage of the (potential) winnings.
In cash games for example, a player can be staked for a certain number of hands. In regard to tournaments, a player can be staked for individual MTTs or a certain number of SnGs for example.The purpose of a staking from the stakee's point of view is to reduce the variance in MTTs for example, or to allow him to play limits for which he does not have the necessary bankroll according to his BRM. For the staker, it's a kind of investment from which he is hoping to earn a high yield.