DS1, like other spacecraft, communicates with Earth through the use of radio waves. Radio waves are sent back and forth between Earth and spacecraft. Communications can take many forms. They can be commands for what the spacecraft should do, information about the basic health of the spacecraft, or information collected during space flight. In the case of DS1, most communications are from DS1 to Earth---the downlink being mostly information about how the new technology on DS1 is performing in space.
How does NASA communicate with spacecraft?
How does DS1's communications system work?