Staking generally means that one or more people provide money to a player to play with. In the best case, the staker gets back his “stake” and an agreed share of the (potential) profit.
There are a variety of modalities possible. Staking is available in both tournaments and cash games.
For example, in cashgame, a player can be cased for a certain number of hands, in tournaments for individual MTTs, or a number of SnGs.
Stake’s purpose, for example, is to reduce the variance of MTTs or to be able to play limits that his own BRM does not yield. For the Staker it is a form of investment in which he promises a return.