The reason you might want to hide your IP address is if you want to stay anonymous on the Internet, so that sites you visit won't be able to trace you. Law enforcement officers or detectives who don't want their IP address left in the log of the Websites they visit in the course of their work are examples of people who need to cover their tracks as they gather sensitive information. You have a number of options, with the most usual one being the use of a proxy server. When you connect with a proxy server and ask for a resource available on another server, the proxy connects with the other server or brings up what you ask for from a cache. Some proxy servers don't pass along your IP address, but some will. Some charge for their services and some don't.
A transparent proxy does pass along your IP address, so it won't do a very good job of hiding your identity. An anonymous proxy doesn't pass along your IP address, but it does identify itself as a proxy server when connecting with the other server, while a distorting proxy provides an incorrect IP address. A high anonymity proxy doesn't pass along your IP address and it doesn't identify itself as a proxy server.
If you install proxy server software on your computer, you'll either pay a one-time fee or pay with a monthly subscription fee, but you'll find that it works faster than the above browser-configured proxy servers. Finally, VPN (virtual private networks) guard your data and identity over the Internet. They're used by people who want to reach Websites that are only available to those living in a certain country and come in handy if you're traveling and want to still access Websites in your country of origin, or if you live in a country with high levels of censorship, such as China, Iran or North Korea.