When I first got my iPhone, my boyfriend set it up with his iTunes and iCloud account because I didn't have one of my own. We recently broke up, and he then changed the password for his Apple ID and iCloud account so I couldn't download any more music and/or apps. I have a lot of things on my phone that I really don't want to lose. How can I back my iPhone up without losing anything on it especially the music and contacts I have?
You can do this by signing out of that iCloud account. Open the Settings.app and go to iCloud. Then select "Delete Account". Not to worry, this doesn't delete any information from your device, as long as you choose "Keep on iPhone" after signing out of that iCloud account. It will display a prompt after you select "Delete Account", and give you the option of keeping iCloud contacts, calendars, and such on your phone. Like I just mentioned, you will want to select "Keep on iPhone" when facing this prompt. After signing out of iCloud, create an Apple ID (there should be a way on the iPhone itself), and sign into it.
As a side note, unfortunately, you will likely have to purchase any music, or apps, again, that were purchased on his Apple ID. The apps that were purchased with his Apple ID will continue to function, but you will no longer be able to update them, and consequently, the need will likely arise that you will need to purchase them again.