The quick answer? Rooting is 'hacking' your phone into granting you access to the most basic level of files (the root folder, thus, the term 'rooting').
Basically it opens the gates of your device and lets you write, read, erase, re-write and modify EVERYTHING in your phone. That's the cool part, the scary part, and the dangerous part. In the right hands, rooting can be fun. In the wrong ones (be it someone who doesn't know what they're doing or someone who maliciously knows exactly what they're doing), it can wreck havoc onto your device, potentially rendering it unusable if you were to delete the wrong file or re-format the wrong partition.