The simpliest, and earliest, form of PTO is the transmission PTO. The PTO shaft is directly connected to the tractor's transmission. The PTO is only working when the tractor's clutch is relased, so if you take the tractor out of gear while slowing down the PTO will stop working. This is a disadvantage in applications such as mowing.
An overrunning clutch is often needed with a transmission PTO. Without it, the driven equipment (such as mower blades) will put a force on the PTO shaft, and then the transmssion, due to inertia. The equipment will "drive" the tractor, and you will still move after using the tractor's transmission clutch. An overrunning clutch prevents this from happening by allowing the PTO shaft to freely spin in one direction. In more recent models, this is built into the tractor. In older tractors, it is an extra piece of equipment mounted on the PTO shaft.
Live (two-stage clutch)
A live PTO works with the use of a two-stage clutch. Pressing the clutch half-way will disengage the transmission while pressing it fully will disengage the transmission and the PTO. This allows the operator to slow down or change gears while the PTO is still operating.
An independent PTO means that the PTO shaft is controlled with a separate clutch. As with a live PTO, this allows for full control over the tractor while separately controlling the PTO. There are two major types of independent PTO; mechanical and hydraulic. A mechanical-independent PTO uses a separate on-off selector, in addition to the PTO control lever. Often the tractor must be stopped or off to change this selector position. A hydraulic-independent PTO uses a single selector.