Cable internet connections are by far the fastest connections of these 3 types. They can reach speeds of 20-30 mbps. They are an "always on" connection that does not require any sort of dialing in. It is not connected to the phone system but is connected to a local cable TV/internet network. The main disadvantage to this type of connection is that, due to its design, speeds can vary. Cable networks involve "clusters" of users that use the same cable lines to draw their connection. When numerous users are connected at the same time, speeds can slow dramatically.
DSL internet connections are the next fastest connection available. These connections are usually in the speed range of 3-5 mbps (much slower than cable connections). These connections are ran through your local phone companies lines. Even though they are ran through telephone lines, they do not require any "dialing in" and are an "always on connection". The main benefit of DSL over cable is that the speeds are constant because each line is dedicated to each user.
Dial up connections are the slowest of all three. These connections are VERY slow in comparison to DSL/Cable with speeds at a fraction of DSL/Cable (56 kbps MAX). These connections are ran through the phone lines and require dialing a number through a computer modem to gain a connection. Most rural areas are stuck using dial up internet connections due to phone and cable companies not updating their systems in those areas.