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: 
The answers so far focus on the algorithm itself, I have a few social economic thoughts to add.

Let's assume Bitcoin is massively popular and indeed becomes THE global go-to currency, at this point this and similar questions become (very) relevant.

What happens in maturing industries is that through commoditization and mergers smaller and smaller numbers of players remain. Through scale advantages this small number of players will be able to provide services at lower cost and squeeze out smaller players. I see little reason the industry of Bitcoin transaction processing will be exempt from this general rule.

Next, we cannot foresee every aspect of the future, even though the Bitcoin designers did a terrific job there will be situations that will call for changes to the system. For example there might be a call from the people to stop child porn networks, to stop capital shelters for the rich, to stop overly profitable and powerful corporations,... etcetera, you name it. Whether justified or not, the people will demand for changes, not necessarily a villain government individual, the people. 

Since there is only a small number of players it is actually possible to regulate the industry. For example the regulation could be that only payments with a traceable account number will be processed, or only payments with attached fees that include a portion for tax. 

I would think the government could even demand changes to the core of the algorithm. Preventing, for example, "non-certified" players to enter, thereby further establishing the power of the existing payment processors.

The newly elected monopolists will then, in the final phase of capitalism self-destruction slowly but steadily raise their processing prices, eventually driving customers away and causing the Bitcoin to never reach the deflationary status many proponents and early investors claim it will have.

And let's just hope it ends this way, a forking scenario from this could be that the Bitcoin reaches "too big to fail" status, and the people demand further regulation (of processing fees, mining speed caps, etc). We will all keep paying a premium on the existence of the currency, just for the sake of stability and the fear for disruption of the status quo. Just like with today's currencies.

I'm not trying to be skeptical, I'm actually very hopeful the crypto currencies are going to help with globalization and advance humanity. As a deflationary currency to "easily" save for your (early) retirement I am not so sure. As a transaction system probably in some way.

Maybe we don't actually need a "currency" maybe all we need is a transaction. Maybe there can be a super layer on top of multiple competing crypto currencies that quickly and automatically switches your money back and forth between the best suitable mix of currencies and investment funds. After all what you really care about is how your salary is exchanged into goods and future promises.
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)