There is no TextEdit equivalent for the iPhone or iPad, therefor you can not sync directly between the Mac and iOS devices using iCloud. What could it be used for? Well, you could sync between Macs.
I have two to three-ish possibilities for you.
First, if you really want to use TextEdit, don't save the documents to iCloud. Instead get (if you don't have it) a DropBox account and set it up on your Mac. Once you do there will be a local folder on the Mac that automatically syncs with DropBox online. Save your text documents there (probably to a subfolder). There are literally dozens of text editors for the iPad that will sync with DropBox. Box.com is another online service that you can do this with, though only about half as many iPad apps sync with it.
Oh, yes, you use GoogleDrive this way as well. Or you could just use GoogleDrive (includes Google Docs) on both the Mac and iPad. I don't care much for it, but others swear by this option.
Second, use the Notes app. This already exists on both iPad and Mac, and will sync via the cloud. If your text needs are simple, it will do. I don't recommend using it for long term storage of text documents. When something changes on the iPad or Mac, Notes updates everywhere fairly quickly. That makes it easy to lose a document.
Then there is Evernote. I really love this app, and it's available just about everywhere. The text editing features are decent, and it will do a lot more than just create text notes. If you pay for the premium account ($5 or less a month) you can have all your notes local on the iPad, and access to note history (handy for those massive editing mistakes.
If you haven't guessed, I mostly use Evernote.
However I do have a couple of dedicated text editors for those times when Evernote isn't enough, or just the wrong environment.
Almost a word processor, and with massive export possibilities, Textilus. It will work with DropBox. Depending on how you save the documents they will work fine in TextEdit.
My other favorite is Daedalus. It's an unusual environment that uses stacks of pages, instead of the usual books and folders metaphor. But it's a very clean and distraction free text editor. It can by synced with either DropBox or Box.com. Or if you're willing to spring for it you can get Ulysses from the Mac App Store. It will sync directly with Daedalus. I don't have it, but it comes well recommended by some tech writers/bloggers I follow.
If you ever need a good word processor with layout controls, Pages is your best bet. If you're already on a Mac, then there is no better cross-device personal word processor for OS X and iOS. Purely my opinion, of course.