Within my app, how can I make it check if there is an update available in the Mac App Store, and tell the user about this?
As an example, Sparrow does this.
Charcoal Design has an open source component that does that: iVersion. But it requires you to add a file in your server for your application to read. Whenever you update the version, just change the information in your server, and iVersion will show the user that a new version is available.
You could also scrape Apple's servers to read the version of your app, but there is chance that your app may be rejected for doing that.