Turnkey PCB Assembly is the complete PCB Assembly process which includes the fabrication of the board, procurement of components to be placed on the board, the PCB assembly process and other testing or quality control processes that are required to give customers the final assembled printed circuit board.
So, when a customer asks for turnkey PCB assembly, this means that they are looking to get a final PCB board and expect the PCB company to provide all the services required from start to finish.
The alternative to turnkey PCB assembly is to get each step in the PCB assembly process done separately or by different vendors. For example, one vendor would fabricate the board, then someone would be responsible for component procurement, and another company for PCB Assembly and testing of the final board. All this requires a significant amount of coordination.