In short, ADSL is a type of DSL.
DSL stands for Digital Subscriber Line and is the parent word to describe the technology that allows the transmission of data over the copper line originally installed in the ground to connect you to your local telephone exchange in order to transmit your phone calls. This data represents your connection to the internet.
ADSL describes a type of DSL and the 'A' stands for Asymmetric. The asymmetric puts more emphasis on the download (as this is where people need the fastest speeds), providing greater download speeds then upload speeds. Further enhancements to this technology have allowed download speeds to ramp up to 24Mbps, or 16 times faster than DSL which means your internet browsing is faster, your emails download and open quicker.