I know this was from a few days ago, but since you didn't get answers to your last questions....
When you get the message that purchased apps haven't transferred (or music, or anything else like that), it means that you "purchased" (it's considered a purchase, even if it was free) apps (music, etc.) on the iPad, and iTunes on your computer does not have a copy of the app.
Whether or not you transfer them is up to you. Transferring them means you use up more space on your computer - you can't run the apps on your computer, so they are only stored there as a way to help restore your iPad. But, if you do back them up, it makes it a bit easier for you to restore your iPad if for some reason you need to wipe it or it has issues.
If you want to transfer them, just say "yes" to transferring them when the computer asks if you want to. And "no" if you don't want to.
The apps don't take up a whole lot of room, so transferring isn't a bad idea - but it can take a while if you have purchased a lot on the iPad that are not on your computer.
I'm hoping you have resolved your problem by now, but if you haven't...
When you sync your iPad, make sure you do a backup as well. If you look at the summary screen in iTunes while your iPad is connected, you should see the option to back up. I recommend connecting with a cable - it will go a lot faster than doing it wirelessly. If you have any problems finding how to do a backup, check this article from Apple to help you:
As long as the iPad is backed up, you should be able to reset it and restore it. I wouldn't do that without trying everything else first, but as long as you've done a backup, it shouldn't be a big deal.
The hard reboot should take care of most issues though. And yes, to do a hard reboot, hold down both buttons (on the top and on the front) at the same time, ignore the slider to turn off, and keep holding until it actually shuts off. If it doesn't restart on its own after that, hold down the button on top a few seconds until the Apple logo appears. That will restart it.
If you have any more questions, just ask. I don't check frequently, so you might want to PM me just to make sure I notice you have another question.
[This message edited by osxgirl at 5:32 PM, August 15th (Thursday)]