There are a bunch of things you can try. Start with a soft-reset (hold both sleep and home buttons in until the Apple logo shows). Then try logging out of iTunes / App Store - in "Settings" as well as the app front-end before trying to download again.
If those don't work next step is to try updating your device's iOS version from a computer instead of on the device itself.
And lastly, if all the above doesn't graft, you'll need to do a factory-reset ("restore") - AFTER you've made a new backup to your computer. After the restore try to download an app before restoring the backup to determine if the problem is solved - bcause restoring the backup MAY restore the problem to your device too! This way if it DOES happen you know where the problem is. If that works then restore the device backup. Hopefully you can still download / update after the backup is restored. If not then you'll need to do a factory-reset AGAIN, then slowly restore content to the device.
Good luck! I just had to go through precisely the above myself for a client's device. Not fun. But problem solved eventually - that's the important thing!
~Moi (aka Menageryl)