The 6569 VIC-II chip is the graphics chip that made the C64 famous, supporting different graphics modes at different resolutions as well as sprites in a way no other machine at the time could.
Although the VIC-II doesn't very often stop working, it can cause faults such as:

Blank screen of a random colour

Blank screen with no sync (monitor will say "No Picture").

Black screen (also a symptom of many other things).