Assignment is the other side of option exercise. When option holder exercises the option, option writer is assigned – receives an assignment notice and is obligated to take the other side of the exercise transaction – sell the underlying at strike price if short a call, or buy the underlying if short a put.
With US exchange traded options, exercise and assignment is processed by the Options Clearing Corporation (OCC), which randomly chooses and assigns one of option writers (all those with an open short position in the particular option contract) for each exercise.