You need a Wireless Broadband Router and, if your broadband is ADSL via the phone line, this should be a *combined* "Router & ADSL Modem". For Cable Broadband (NTL & Telewest) a "stand-alone router, often described as a "Cable/DSL Router", is used.
Any computer that you want to connect to the router "wirelessly" with needs to have a Wireless Network Adapter installed. For a desktop PC this would be a PCI adapter or USB, for a laptop PCMCIA or USB. (I don't know what you would do with an Xbox.)
You can, of course, also connect to a router with a conventional "wired" ethernet adapter.