Do note there is a 'difference'.
1) If you purchased off itune or what is known as 'in store purchase' then it is fine, you just need launch the App Store App and navigate to the Purchased icon on the bottom right, it will load all the purchase app registered under your Apple ID, click on those and you will download it free.
2) If you have bought some apps 'in app purchase'. Lets say you download a free trial app (or a purchased app) and somewhere while using the App it has some new features or levels and there is a button to 'purchase in app', you click it and you bought it. Sometimes this purchase is NOT recorded back to the Apple record (it's still on your iphone/ipad) The only way to do it is to connect to the computer, inside itune and click the transfer purchases which will transfer that information to itune and then to Apple record.
I had 2 occasions when my young son was playing games apps he likes (shrek cart) and he purchased it, but there was a flaw on the programming and it can NEVER sync back to the itune, the purchase feature is still stuck on my iphone. So I reset to factory and tried to redownload the app but found that I had to purchase it again! Sending an email to the Apple support solved the problem (they removed the purhcase). But my lesson is a) never to leave my password saved by default, b) never to forget to sync my in-app purchase again.
3) This is related to item 2 above, when I reset my Iphone, and I went to the Appstore to redownload the app which I already purchased, there was a bug from the App developer. The same App, but different version charged me AGAIN for the same app. I thought I just clicked the 'purchase' button, and somehow Apple will know my record, that's wrong. The first app I bought was version 1, and the same app I tried to redownload was version 1.6 (or something) and the App developer has the audacity to charge this purchase again. So again I send an email to Apple support, and they checked that yup, I had bought the same App twice!, they did a refund for the 2nd purchase.
lesson learn is that to do it properly via item 1 as shown above. don't search from the categories and think you are downloading the same App that you have purchased, it may be different.