It depends on the app.
I try to browse for apps on my iPad, because I can easily tell if they're compatible for both or not. If there's a + in the upper left corner of the box containing the price, the app works on both devices and doesn't require a repurchase. I bought the permium version of Docs to Go on my iPad and was able to install it on my iPhone without paying again. Quickoffice was not only more expensive, but it has separate versions for the iPad and iPhone, which factored into my decision of which one to get.
In the App Store on your iPad, you can go to the purchased tab and see all the apps you've ever downloaded. In the top right there's a toggle for iPad/iPhone. Select iPad and it'll show you all the apps you've already downloaded that are compatible.