Whenever we update one of our games in the Nook store, it may show up for you as an notification in your top bar, and give you the option to download it.
Sometimes it might miss an update (perhaps if you dismissed a previous notification), but you can always do a check for any App updates like this:
1 - Go to your Apps page by pressing your Nook n button and then tapping the Apps icon,
2 - Tap the 'menu' icon in the bottom left hand corner
3 - Select 'Check For Updates'
It will check with the Nook server and if there are any updates available it will show them to you. It might take a few seconds to do this, especially if you have quite a few apps already installed.
You can select just the ones you want to update, or 'Update All' to do them all in one go.
4 - When it's done, it will show you that all the apps are up to date, with a screen like this.
You can check which version is installed by looking within the 'About' screen in the app itself, or by checking the Settings app, and then browsing for the app you're interested in.
If you've just run an update then you should have the latest version installed, however you can check to see that the version number matches the 'latest version' shown in the Nook store.