How to use AT&T iPhone for Verizon?

I got an iPhone 5 for Christmas so I have my iPhone 4 to my boyfriend. I had AT&T but he has Verizon. How can we switch it? Will jail breaking it make it able to be switched?

No it can't be switched since it came from att and is going to verizon

Verizon and AT&T use two different types of technology. Their mobiles can't read the other company's towers' signals. This may change in a few years once all the carriers' LTE buildouts are complete.

AT&T iPhone 4 is a GSM phone (takes SIM) and Verizon is CDMA (doesn't have SIM) so they're incompatible.

Nope.

All AT&T devices use GSM.

Verizon is a CDMA provider.

There's no way to make a GSM device work on CDMA networks.

Verizon phones are technically different from ATT phones. Jailbreaking will let you downloading extra applications, but it will not help you to change the phone's technical specifications.