App updates, like purchase and installing of new apps, is handled directly by Apple.
On the iPhone/iPod touch, app updates are handled by the App Store application.
If updates are available, the number of updates is displayed on a badge on the App Store app. After launching the App Store app, you can select single or all available updates in the "Updates" tab.
Alternatively, you can download available updates also in iTunes on your computer: Are updates available, iTunes displays the number of available updates next to "Apps" in the navigation sidebar:
Select the Apps section and click on the "Check for Updates" link in the lower, right corner of the window.