Do a search on the archives and you can find about 4 hours of reading on live/independent PTO's.
The Power Take Off (PTO) shaft extends out the rear of the tractor and turns at a nominal 540 rpm. It is used to power any external attachments to the tractor such as a mower, tiller, or a lot of different agricultural implements such as a sprigger, bailer, etc. It is a very powerful shaft and requires a lot of respect to use it. Getting a shoelace tied up in one can lose a foot and getting long hair caught in one can lose a scalp. The protective covers are there for a reason when they are in use.
Independent PTO means that the PTO will keep running when the tractor is stopped if it is engaged. Main use is for agricultural implements, generators, or a log splitter (would be hard to split wood only while the tractor was moving), etc.
Some PTO shafts turn at 540 rpm (when tractor engine speed is correctly set) and that is the most common usage. Others turn dependent on ground speed (very useful if you are running a corn seeder or fertilizer spreader). Some tractors have a lever to select one or the other.
PTO's are engaged on older tractors and some newer tractors through a two speed clutch for a gear tractor. The clutch first engages the pto shaft and then when you let it out a little further moves the tractor. Obviously pushing the clutch all the way in will stop the pto and cause loss of hydraulic pressure on some tractors as well.
A lot more depends on the implementation.
Most of this is kinda meaninless if you buy a hydro transmission as the pto is generally independent.
Collar shift is a John Deere term that means it has a manual gearbox.
Synchro shift means that the transmission is probably at least synchronized in the forward and reverse gear for loader operation.
The main differences in the gear transmissions is that fully synchronized transmissions are now available in a lot of models. Some even have synchronized forward/reverse levers to make shuttle shifting with a loader or just moving forward to reverse much easier with a mower or other implement.
