Question: Suppose organization X has 51% of the hash power for a period of 1 week. In this week, what exactly can and can't X do?

Answer: 
In theory, this attacker owns enough computing power that they could execute a "double spend" attack. They could spend coins in one place, allow the coins to enter the block chain as normal until the required confirmations are met, then fire up their 51% of the miners to craft a fraudulent fork of the block chain in which those coins were never spent, allowing them to re-spend the coins. This could theoretically be repeated for as long as the attacker maintained control of 51% or more of the hashrate.

Realistically, 51% is only the point at which this becomes *possible* not the point at which it becomes *likely* or *easy*. An attacker would probably need something like 65% to actually execute such an attack.
I suspect he might be able to mine more than 51% of the blocks. See [Can someone with 51% computing power earn more than he deserves?](https://bitcoin.stackexchange.com/questions/1475/can-someone-with-51-computing-power-earn-more-than-he-deserves)