Before this topic gets flooded with a lot of incomplete information:
Any wireless-device knows basically 2 modes (there are a lot more configurations possible, but these are the most popular):
-Access Point Mode
-Client (Managed/Ad-Hoc) Mode.
When in Access Point Mode, a device will accept connections from clients (who can authenticate with it in a few different ways), so they can access the wired network (either routed or not, OSI-level 2 or 3).
When in Client Mode, a device will try to connect with an Access Point (also called Managed or Infrastructure). A Client can also be reconfigured to connect with another client (Ad-Hoc).
Connecting two devices in Access Point Mode isn't really a standard. Most of the time, you will need two AP's from the same manufacturer. Anyway, it takes a lot of configuring for a novice.
The problem with hotels is that they will have Access Points and that AP of yours probably can't bridge with them. A solution might be to run Linux on the device (which can bridge by putting an AP-device in client mode, thereby making everything transparant), but this is a big quest.
My solution: buy a 2nd-hand laptop (which can work with Windows XP+SP2) and a wireless PCMCIA-card. I wouldn't drag my Xbox along just for checking mail/the internet. Games might be interesting, but then again...