It depends on what modules do you install with it. If you don't install/use any extra modules, there's 3 processes running for the Avast; GUI process (that shows the system tray icon), automatic update process and the main Avast service process. On my system, currently those 3 processes takes about 13Mb of ram. Update process only takes 292Kb.
Note: Network Shield module doesn't run as a separate process. That's the only extra module i use with Avast.