Don't get discouraged! You're doing fine
I think most people starting an exercise program tend to expect too much too fast when it comes to building up endurance - then they get discouraged and quit because they can't do what they see others doing. Those people you see on the elliptical for over an hour have probably been doing it for at least half a year, possibly many years.
I started a little over a year ago, and I can go for an hour, but I think it was a good six months of doing it five times a week before I could manage it. It took a month before I could finish my first half hour, and that was done with half of it at a slooooow speed.
Keep pushing yourself, but don't overdo it - you want to be able to climb back on there the next day (or the day after) and do it again. Do what you can easily do, and add 1 extra minute as your goal (don't aim for 5 extra minutes - that's only going to discourage you). When that's easy, add another extra minute. Keep repeating the extra minute, and you'll be able to do a half an hour sooner than you'd think.
As others have mentioned, slow down, lower the resistance or incline, and go by what your heart rate says. I know how much you must want to get fit NOW, while you still have the motivation to get over the hump, but it is better to slow down and let your body pick the pace that it can handle right now. Think about how far you have come in two weeks - you can *do* 10 minutes, even if it is hard. In another two weeks, you will be able to do more - even if you go slow with your adding on minutes, you'll be able to do that 10 minutes easy-peasy.
BP