802.11g-compatible gear is the current market leader.
802.11n has greater bandwidth, but is not common and may not yet have had its standards nailed down (anybody know for sure?)
802.11b is an older standard and 802.11a is a short-range alternative to 802.11a (more to it than that, but that's close enough)
If it were me, today, I would run with 802.11g gear.
Exactly what hardware you'll need depends on how you get the fat pipe (broadband) into your house.
Assuming you have a fat pipe, most folks go with a combo wired/wireless router. I use older Linksys WRT54G hardware I find on ebay for several reasons (google "ddwrt").
If the router is near a desktop, please do use CAT5 or CAT6 to connect.
The other very important part of the deal is directional antennas. I like to have my router in a corner of my house with a 90deg, 14db directional antenna bathing my home in 802.11g goodness. Stock antennas are usually 2db and have a much more limited range.
Last, any computer must have a wireless card / antenna of some type.