Jailbreaking is when you hack the operating system on your smartphone or device (typically an iPhone) to override the operating system or service provider. It's not advised, as it can void a contract, get you in a lot of trouble with a service provider or manufacturer, and (sometimes) ruin the smartphone or device.
Or, from about.com: "Jailbreaking is a way of freeing it from the limitations imposed on it by AT&T and Apple. Doing so is risky, though. It can void your warranty..."
When iPhones were first released, and only available on AT&T, many consumers jailbroke the phone and ran it on Verizon. (This did not please Apple, AT&T, or Verizon.)
Some users wanted to access parts of the system that were not otherwise available. Jailbreaking allowed the users to access all parts of the system.
If you attempt this (and there are plenty of resources out there), be careful.
What Does It Mean to Jailbreak an iPhone? , About.com
What is Jailbreaking?, Apple iPhone School