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.
And then there is the denial of service possibility of suddenly withdrawing from the service, taking the necessary computing resources away to continue to solve blocks every ten minutes until the difficulty is adjusted down again (which could take a long time if there is only a block every day for example).

Of course, for that one would need much more than 51% of hash power.