From your iPhone/iPod/iPad:

Step one: Open the app store app

Step 2: Tap featured on the bottom tab bar, and if it isn’t already selected, tap either “New” or “What’s Hot” on the top tool bar.  Then scroll to the bottom, and tap “Redeem”.

Step 3: Enter your promo code!

One you have submitted your code, the app will automatically download to your iOS device.  The app will usually install to the first available space you have on the device OTHER than the first page.  So, if you have an open spot on page 2, then it will be there, if page 2 is full, then it will be on 3, etc etc.

If you entered a code from an iTunes gift card instead of a promo code from a developer, then the sum of that giftcard will be applied to your iTunes account, which you can then use to buy some cool apps.

From iTunes

If you happen to be at your computer with promo code (or iTunes gift card) in hand, you can also enter it in iTunes.

Step 1: Open iTunes, and click “Store”

Step 2: Find the “Quick Links” section on the right hand side of the iTunes app, and click “Redeem”

Step 3: Enter you code, and click Redeem

This will start downloading the app if it was a promo code, or apply the cash value to your iTunes account if it was a gift card.  The only drawback about doing it this way for promo codes, is that you have to sync your device with iTunes before it will be installed on your device, which kinda sucks.  The easiest way by far for promo codes is to enter the code on your device.