The plug in kind will go out if the power goes out. A quick solution to the treadmill problem is one of the battery backup units for computers. The cheapest one you can find will do the trick, so if the power goes out you won't lose the information about your workout. I picked up a few at Best Buy around the holidays last year for about $5 each after a rebate.
An elliptical is different than a treadmill though. A treadmill gets harder to move when the power goes out, and elliptical is the opposite, it gets easier, so you wouldn't have to worry about anything happening if it lost power (or your workout completed.) It is more like a stair climber in that regard.
Basically the power part controls the tension on the flywheel and the LED display. You generate tons of watts of energy on those things, so it takes some juice to provide resistance (or hold the spring in place to provide it!)
http://www.diamondbackfitness.com/depts.asp?deptid=1 is a link to the one I got. I did a lot of research before buying, and found that to be the best for what I wanted. Make sure you try one out in the store before buying, and don't be afraid to give it an honest workout in the store. I got some weird looks in Sears, but more people in passing said, "that's a good idea, if I'd have done that I wouldn't have the paperweight of a __________ in my garage."