In Windows 8 the process for getting updates to any apps you have downloaded from the Windows Store will be somewhat similar to the experience on a Windows Phone handset. The Windows Store tile will get a small notification indicator, a numerical number, in the lower right corner of the tile which indicates how many apps have updates waiting.
Another apparent step in getting your attention that apps need to be updated is an indication on the apps own tile. That indication is a small white circle with a black X in it. You can see both the Windows Store update indicator and the app tile indicator in the screenshot below:
If you click on the app tile you will be taken to the apps information screen in the Windows Store but it will be overlaid with a banner indicating the app needs to be repaired. You are then offered a chance to repair/update it from the banner or you can cancel the notice.
If you click on the Windows Store tile you will enter the Windows Store and in the upper right hand corner will be a text indicator that updates are waiting for your installed apps. Click on that text and you will enter the App updates area where you can select which apps to update. Clicking the Install button on the bottom right will update your selected apps.
If you are curious what has been updated with your app then you need to visit its info page in the Windows Store and click on the Details tab. The release/update details should be at the top.