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It's an ongoing ability. If it is discarded its effect will persist until the end of the phase it was discarded in. Unlike Woes it does not specify "for one round" so if you do not interact with it it will negate the same target indefinitely even over multiple rounds.
So if my opponent plays the three woes on his turn to negate my artifact when it's my turn can I turn it off and back on to use it again or is it still negated?
Face down artifacts are "in territory" but they are not in play. A card that discards a card "in opponent's territory" could discard a face down artifact.
That was actually an incorrect statement as face-down artifacts are now considered in play.
If you turn it down then back up again what you've actually signaled is that you're leaving it active. You'd have to actually deactivate it for a round but then after that you could use it again.
On a related note to Kevin, how does this interact with Temples? Say I have Ark of the Covenant active in my artifact pile and it gets negated by Image of the Beast. I can't underdeck Image (an idol) if I leave Ark up. What if I move it to a Temple? Does that count as de- and re-activating, a la U&T, or is it just always active?
If you were to move the Ark's location to a temple instead of your art pile it won't have any affect on it still being negated, deactivated or activated. You're simply changing its location.