The posted answers to your question aren't exactly correct.
With the Xbox 360, it breaks down like this:
A NTSC unit will not output a PAL signal. In standard definition, it will always output 480, which is NTSC. Most newer TVs in Europe are multi-standard and will accept a NTSC signal.
If you output in HD (720p or higher) then there is no difference between NTSC and PAL. Those designations really only apply to SD signals. 720p is 720p no matter where you are located.
As far as games are concerned, your console will play any games that are region coded for North America as well as any games without a region code. Most Microsoft first party games are region free and do not have a region lock. With third party titles, the
region lock is decided on by the publisher. If a game is region free, it will play (and output properly) on your console. Unfortunately, the only way to confirm if a game is region free or not is to look it up online or try it (I suggest renting) as this is
not something that is publicized on the game boxes.
Finally, there is Xbox Live Arcade. Your XBL account is probably set to the US region. Anything you've downloaded will still work. However, going forward, you may run into some games that cannot be downloaded from NA XBL while you are physically located
in Europe. This is due to licensing. Any MSP cards you buy in Europe will not work on your US account (you would need to get US cards). XBL Gold time bought in Europe *may* work on your US account, but again, you would be better off buying subscription time
from a US seller.
If you are going to be in Europe for the long haul, you may want to use the online tool to change your XBL account region from North America to your current location in Europe. After using the migration tool, your account will connect to the local version
of Xbox Live and locally purchased MSP and XBLG time cards will work. However, American cards will not.
Hopefully this helps answer some of your questions.