Staking is a process that utilizes the Proof of Stake? system. There are many parts and validating criteria but one feature of this system allows for those deemed trusted holders of coins within a client wallet to be rewarded by Staking those coins, this means the more coin weight they have held in a supporting client wallet, and, if left untouched for a certain amount of time or set blocks that have passed, they are then eligible and can begin to compete for the next block and thus rewarded in the form of a Stake.
This is vastly different from the Proof of Work? system in which power-hungry computational hardware has to solve computationally intense problems to validate transactions on the blockchain and then go on to create new blocks, the miners? are incentivized for doing this by getting a mining reward, if successful - as with both these protocols the main goal is to create a truly decentralized and spread out system and deter cyber-attacks such as a Distributed denial-of-service attack DDoS? or even platform specific attack such as the recent fake stake exploit.