When you have a large media library, you can't fit every album and movie on your iPhone at the same time, but as long as you use the same Apple ID, iTunes lets you re-download old purchases for free. You don't even need to use your computer to see your old purchases: the iTunes Store app in iOS 8 has a complete list that shows you either your entire library or the specific items that aren't already on your iPhone.
Step 1: Sign In
Run the iTunes Store app on your iPhone. Scroll to the bottom of the page, tap Sign In and log in with your Apple ID. If you're already signed in to the correct account, you don't need to do anything. On the other hand, if you're signed in but want to check purchases on a different account, tap Apple ID and choose Sign Out.
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Step 2: Open Purchases
Open the More tab in the iTunes Store app and tap Purchased.
Step 3: Choose a Media Type
Pick either Music, Movies or TV Shows.
Step 4: Browse Purchases
Choose either All, to see all iTunes purchases, or Not on This iPhone, to limit the list to purchases that aren't currently downloaded or synced to the phone.
You can't search or change the sorting of the purchase list, but tap any item to see more details. For example, in your music purchases, tap an artist's name to see albums and songs you've bought. Tap Recent Purchases to see the 250 most recent purchases. After opening a particular item that isn't currently on your iPhone, tap the cloud icon to re-download the purchase for free.