What determines your IP address? Internet provider, computer or what?
Your internet provider. They usually have a bucket full of IP addresses and hand them out as your equipment requests them. Usually your modem requests the IP address. Say you have a wireless router hooked into a modem. The modem gets the IP address which is how you connect to the internet, and your wireless router takes that connection and splits it however many ways as you have devices connected to it, they all share the same IP address.
Can it be changed?
Yes. If you unplug your modem, your IP gets put back in the bucket. Then when you plug the modem back in, it will likely get a new IP address from the bucket. If you do it really quickly, you might get the same IP address. If you unplug your modem and wait overnight, you will probably get a different one. In other cases, such as here at my office, there is no modem, I connect directly to the network and get my IP address from a DHCP server which has its own bucket of IPs. I can open a command window and type "ipconfig /release" and then "ipconfig /renew" and get a new IP address.
If I have an IP address at my home now, which I do obviously. If I buy a new computer will it have the same IP address (I believe it will)? Also what about cell phones? If I get a new cell phone will I now have a new IP address?
If you connect a new device to your wireless router, you shouldn't get a new IP address because the router is just splitting the Internet conneciton given to it. However, as Chris said, your provider might randomly hand you a new IP address sometimes, it's up to them.
