What does jailbreaking a phone mean?

You get an iPhone from AT&T with a 2 year contract, you are "jailed". Jailbreaking is hacking the iPhone so you can use it with any service anywhere. It also means you can get unapproved apps that can infect the phone and other wonderful hassles. People just want to be free no matter what the cost.
I've seen this for years and years, there are people who like a gadget for the gadget, like Stereo in the 70's and 80's where the equipment was way more important than the actual music. Or computer people who are more interested in specs and performance instead of actually just doing something with it. In the 50's and 60's is was cars that could do 3 times the speed limit that couldn't be run legally anywhere at that speed. They were/are more interested in engine and looks than actually going anywhere. It will always be the way a subset of people become enamored of a thing for it's own sake rather than its original purpose.

It's mostly used when talking about iPhones or iPods. When you first get a brand new iPhone or iPod touch, you do not have root access to the device. Jailbreaking the device gives you root access to it, allowing you to fully customize it to your heart's desire.

People often call it "hacking", but that would only be true if you were the one who made the jailbreak exploit.