the feature you guys are talking about is called the "resume" function (as in, resuming the show where you left off). Some players have it, some dont. Some allow you to set various preferences for the resume option. I have a PS3 & it doesn't default to resume (even if you turn the player off while the movie is playing), but I've never checked to see if it can be turned on because I never really cared one way or the other about the feature.

The DVD player on my computer (I use Windows Media Center as it was the player I liked best of the various ones that came pre installed) has one of the nicer resume features as it can be turned on & off easily, remembers EVERYTHING, and even if you turn it off for a while, when you turn it back on, it still remembers all the disc locations from before turning it off.