How does the iPad auto-correct work exactly? When I type in something incorrectly, or if I'm in the middle of typing, the iPad sometimes shows up an auto-correct suggestion. If I click on the suggestion to choose it, it disappears. And then, when I type the word again, the auto-correct  doesn't appear.

How do I get the iPad to actually respond to my auto-correct choice? I tried Googling, but it didn't turn up any results.
AutoCorrect will automatically replace the word unless you cancel it manually by tapping it. (Note the small × sign in the suggestion bubble indicating that tapping it will cancel the suggestion.)

So, as soon as an AutoCorrect suggestion appears that you want to use, just hit Space (or insert a punctuation mark) and continue typing.