If the two devices both use the same primary app market, then yes you can! (Note that this applies only to things purchased with real money and that can't be used up. Specifically, coins and things bought with coins cannot be transferred.) Please go to the page appropriate for your situation:
- iTunes (purchases made for the iPhone, iPad, and iPod Touch)
- Amazon (purchases made for the Kindle Fire family of products)
- Google Play (purchases made for all other Android devices)
If the two devices use different app markets, then unfortunately you cannot. This page explains more about why this is the case.