What Exactly is Jailbreaking?
There are actually a few different definitions of a "jailbreak". Some include having Cydia on your iOS device. Some are having the ability to run tweaks or modified software. The technically correct definition is having access to the root file system. ROOT is a user profile on your device that you are not normally allowed access to. This user is handled by the operating system, and has access to files in the top (root) directory of the filesystem.
So, what's the big deal?
Most jailbreaks come with a package manager, and a way of running tweaks. The most popular of these being the package manager Cydia, and it's injection system, Cydia Substrate. These use the permissions granted by a jailbreak, allow near complete control of your iOS device. You can change things as simple as application icons, you can completely overwrite the functions of each button, or make the operating system unrecognizable. Above all, it lets you increase your device's functionality.
Simple but unique