ADSL means Asymmetric digital subscriber line. Essentially what Asymmetric mean when it comes to the connectivity is that you have one transfer path that is a far greater transfer rate than the opposite, for example, a ADSL connection may have 8mbit down, and 1mbit up, the downstream path is a higher ratio than the downstream. You could say that nearly all internet connections are asymmetric, as nearly all of them, besides fiber optic connections, are higher down stream than upstream.
The term Broadband varies depending on the country that you live in and their communications regulations, historically speaking, broadband used to be anything that was faster than Dial-up, and over time the definition has changed to represent minimum acceptable speeds for internet connectivity in residential and business.