The X-ray data obtained by Chandra is typically stored on board for about 8 hours worth at a time. About three times a day, Chandra communicates with the Deep Space Network, sending its newly acquired data back to Earth while receiving new instructions from the ground.
The Deep Space Network consists of three dishes in Australia, Spain, and California. It is the primary way (think of a cosmic phone line) that Earth maintains contact with missions through the Solar System.