A modem is a piece of hardware that is used to carry analogue signals that contain digital information and then decode that digital information at its destination.
An example of this is a typical modem that converts 1s and 0s into sounds that can be transmitted over a telephone line, and then once it arrives at the computer it is then re-converted into 1s and 0s through a connection, whether it be a USB, Ethernet or network connection.
Modems can be used for DSL connections as well as dial-up connections. The speeds obviously vary depending on the connection as dial-up connections typically are around 56.6kbps whereas DSL is now typically at between 8 and 10 mbps.