lets start with local files. lets say you begin watching a movie on your computer and stop 20 minutes into it and decide yo uneed to have it on your phone. Sync that movie to your phone and it should pick up (conitnue) where you left off on your computer. Then lets say you want to watch it on your appleTV when you get back home, sync the phone to the computer and then begin to watch on your tv and it should pick up again where you left it off on the phone. Same with podcasts, songs, tv shows, etc. I assume it also works for books, but I am barely getting into using my idevices for reading.
Now if you are watching a movie from the cloud on your appleTV and stop halfway, you should be able tos tream from the cloud to your computer and it will begin from where you left off.
that is how it has worked for me, but I do have to say that I dont use match and barely stream from the cloud, so that end of it I have have limited experience in. but it has worked the few times I have tried it.