They are versions of the IEEE wireless standard 802.11 and refer (mostly) to the speed of the cards. Ignore a. b is old hat (11 Mb/.s). Current cards are 802.11g (54 Mb/s) and that is what you should buy. (They are backward compatible with 802.11b.)
No you can't use a router connected to dial-up. You will need to either use a modem (as you do now) or move to broadband if you want to use a router.
t, August 2007