In analog systems; The frequency range a channel is using or cabable of carrying. For example ADSL works because the bandwidth used by the telephone system (POTS) is only about 3KHz, while the copper pair between the subscriber and teleco has approximately 1.5MHz of bandwidth (on an average-length cable run)
Dialup modems go through the normal phone system, so can only use an analog bandwidth of about 3Khz.
In digital systems; The amount of data a channel can carry, usually measured in kbps, Mbps, Gbps, etc..
Dialup modems can manage up to 56kbps under 'ideal' circumstances. ADSL peaks out at about 7Mbps (someone check this?) although Telecom likes to cap it at 128kbps for home users.
(someone else might want to list the bandwidth of other commonly-used media here :)
The 404-haiku I got when starting this page seems strangely appropriate;