Broadband used to be defined UK wide as having a download speed of 512k or above or "10 times the speed of a modem", however, some companies realising that there was no actual
official definition decided to apply the term broadband to slower packages to make them more appealing - NTL were one of the first to do this that I remember but there have been several jumping on that bandwagon, AOL offer a "silver broadband" package that is only 256k. For a "true" broadband speed of 512k or above you're looking at £20+ a month for an always on, limit free* connection. ADSL is just one type of broadband; cable, satellite and wireless connections are others.
Heres a decent explanation! * some broadband packages offer you a cheaper connection but cap your downloads so they either cut you off or charge you more if you go past this limit. Watch out for this and make sure you want it.