Between ATC and the pilots it means the aircraft is authorized to taxi onto the designated runway AND take off (as opposed to "line up and wait", which means taxi onto the end of the runway, but don't take off yet).

The Pilot/Controller Glossary (US Federal Aviation Administration) defines it as "ATC authorization for an aircraft to depart. It is predicated on known traffic and known physical airport conditions."

When the FA or pilots tell the passengers cleared for departure, it's more of a notice to double check seat belts, everything that needs to be stowed is, etc... It's also for the pilots to let the FAs know that an ATC takeoff clearance is immiment and that all FA pre takeoff duties need to be done and the FAs need to be seated.