1. In certain circumstances, a company will find itself in a situation where it wants/needs to fire an employee but owes said employee a favor or wishes to save face for both parties. In this case, before the firing, the employee is given the opportunity to quit instead.
2. Euphemistic manner of referring to a forced resignation.
"John doesn't work here anymore? They didn't mention anything about firing him!"
"No, he got quit since they caught him in the bathroom with two hamsters, a bottle of KY, and the secretary from the front desk."