Here will go mafort's explanation later.
Pioneered by Peercoin, it is an alternative to Proof-of-Work.
Well, let me first start off by clarifying the staking process. Let's say I send 1,600 HYP to my wallet. I will need to let at least 8.8 days pass, and my block will begin "mining" itself, or attempting to stake. Think of this block of 1,600 as a PoW miner whose hash rate is 1,600 multiplied by the amount of time past 8.8 days. So if it is 9.8 days old, my coin weight (or sort of the hash rate equivalent) would be 1,600. If my block is 10.8 days old then the coin weight will be 3200. As the age grows, it is more and more likely to produce a stake reward, because its weight gets larger. Once the block stakes, the network takes that block of 1,600 and calculates the reward. If it took 10.8 days, then it would be rewarded 7.5/365*10.8=355 HYP. The network takes your 1,600 HYP block and completely destroys it. The network then sends you two blocks of 977.5 (totaling 1955). These two blocks are freshly minted, and as such they have 0 age. You will need to wait another 8.8 days for the coins to start mining themselves again.
There is no reason to use a separate wallet address for any of this process. And if you age your blocks longer, then you are missing out on some compound interest. So the "end of the staking cycle" is when you receive the stake reward and are given two new blocks and the old block is destroyed.
Also I need to point out that if you at anytime transfer coins, it starts over the ageing process.