Apple - How can I change the language for automatic spelling correction?

To add to what gentmatt said, in the case you need to keep "Automatic by Language" because you happen to write in different languages, click on "Set Up…" in the list shown for the Spelling option.

On the dialog box that appears, you can change the order for the languages. In my case, this is the order I chose.

Now, I can write organize, and the word is not corrected, nor do I get a suggestion for writing organise. (It happens that organise is shown as incorrect.)

For OS X 10.8:

System Preferences > Languages & Text > Text > Spelling

Or for OS X 10.9 and later:

System Preferences > Keyboard > Text > Spelling

In Mavericks (10.9):

System Preferences -> Keyboard -> Text -> Spelling

or

System Preferences -> Language & Region -> Keyboard Preferences -> Text -> Spelling