ADSL is a type of DSL where the upstream and downstream speeds differ (hence the "asymmetric" part). There are many variations of DSL technology for different types of application (HDSL, SDSL, etc).
ADSL
ADSL is a broadband connection technology which utilises existing copper wired telephone networks. ADSL uses a wide range of frequencies over an existing telephone line to deliver much higher speeds than your standard 56k modem (between 10 and 140 times faster). It is also possible to use your phone while online.
DSL
DSL stands for Digital Subscriber Line. Generally if you have a DSL connection, you are getting it right from the phone company (There are a few ISP's that offer DSL, but the do it in conjuncture with the phone company). DSL is more like the older 'modems' in that they do use the standard copper phone lines that are in your house. DSL requires that there be a special switch installed at your local phone company's main routing station. DSL also has a limitation of a 4 mile radius, in other words you need to live within 4 miles of that switch to be able to use DSL. One of the cool features of DSL is that you do not need to have a 'special' line installed - you can be surfing the internet and talking to someone on the phone all on the same line. This is because DSL uses a high frequency that does not effect the normal use of your phone.
There are two commonly seen 'flavors' of DSL, DSL (also known as SDSL) and ADSL. DSL or SDSL is the more expensive, but offers the most bandwidth. The S in SDSL stands for Symmetric or synchronous, which means that the bandwidth of both the upstream(you sending data) and the downstream(you receiving data) are the same speed. The actual speed you get can vary, depending on the package you sign up for, and how far away from the switch you are. The most common speeds are 256k (256kilobits)/second and 1Mb (1megabit)/second. ADSL (Asynchronous Digital Subscriber Line) is the same as SDSL, but the upstream bandwidth is smaller than the downstream. This version of DSL is generally more affordable than SDSL. Common ADSL speeds are 256kb downstream - 96kb upstream, 1mb downstream - 256kb upstream.