I'd say that broadband defines connection speeds higher than 512 kbps. (The legal recomendation is 2 Mbps, however, this has no real effect)
And broadband is analogue, not digital. So why do so many still call it digital?
Oookaaay...
Well, the DSL-technologies use analogue lines to transfer data. AFAIK, it's possible because they use much higer frequencies than an ordinary modem. Don't quote me on that, though.