Which pairs are auto-inserted depends on the actual language and the bundle that is loaded for it.
The actual character pairs are defined in the bundle preferences with different settings for different languages and contexts. For example, in source code an apostrophe is set up to have itself as a closing character, except for comments and strings. This is achieved using scope selectors.
For example, if you want to enable single quote autocompletion for Plain Text, you will have to edit the preferences of this bundle according to the TextMate manual.
Go to Bundles → Bundle Editor → Show Bundle Editor. In the left pane, select "Text", and "Miscellaneous". Here, add the following line to
smartTypingPairs (note that you have to enter the single quotes within double quotes):
( "'", "'" ),
Close the Bundle Editor, and it should work for every file recognized as Plain Text. If it doesn't work instantaneously, reload the bundles with Bundles → Bundle Editor → Reload Bundles.