How do I get my contacts installed back on my phone after I reset it?

There are many ways to sync contacts, including iCloud, USB sync to an app on your computer, syncing with a non-Apple system that supports contacts such as gmail, yahoo, AOL, Microsoft Exchange, MSN, etc. The way you get them back depends on how you synced them in the first place.

Were your contacts stored in iCloud?

If so, simply logging into your Apple ID on your iPhone should bring them back.

Otherwise, how are you bringing them back?