Buy Once, Download Many
All you need to do is login to that same iTunes account and re-download the app to your new iPhone, iPod touch, or iPad.
If you want to bulk transfer a large amount of apps to a new device, use the ‘Transfer Purchases’ option in iTunes.
There is two exception to this rule, the first being some iOS apps are sold in two versions, say as an iPad “HD” version and a standard iPhone version. The main difference here are the apps resolutions, but most developers will simply bundle the two in a single app and sell a “plus” version (marked by the + sign in the App Store). The other exception are iPad only apps, which are designed to run at a higher screen resolution and can not scale down to the smaller screens of the iPod touch and iPhone, but since they won’t run on smaller screens you won’t have to pay for them again anyway.
I often wrongfully assume what is common technical knowledge, but over the past few weeks I’ve ran into several friends and family buying the same apps again that they already paid for, simply because they want the app on a new device (I’m sure the makers of AngryBirds love them for this). To make things even more complicated, they had setup different iTunes accounts for the different iOS hardware. If you see someone doing this, correct them!