A sentient star was a star that was alive, and could feel, think, react and, in some instances, manipulate certain things around them. Removing part of a sentient star caused it pain.
The Torajii system had a sentient star at its centre. The SS Pentallian used a fusion scoop to mine gases from the star as fuel, and the star reacted by possessing the ship's crew. The Tenth Doctor and Martha Jones followed a distress call and the Doctor realised the the sun was alive. He made Martha dump the fuel with the sun particles and the sun let the ship leave. He made sure the sun had protection, just like any other living organism, after the endeavour. (TV: 42)