It's essentially a port which was designed to do many things very quickly. It had a much greater bitrate than FireWire (probably the port it was designed to replace) and USB, meaning files and data could be transferred at much greater speeds which is great for things like video editing. It could also be daisy chained, so one port could actually have six devices connected to it. It's designed for things like attaching displays, hubs, hard drives, etc. It's a lot more versatile and has a lot greater potential for other uses than HDMI.
Unfortunately, it's never really taken off. Partly because the licensing costs were apparently quite high, low level of adoption from manufactures other than Apple, and the connector it used (Mini DisplayPort) wasn't that widely used on devices other than Apple's either.
However, Thunderbolt 3 uses the same sized connector as USB-C, which is designed to replace the extremely common USB-A. A Thunderbolt 3 port supports USB-C devices (but not vice versa), so it gives us hope that it might finally see wide adoption as it can simply act as a USB port if a TB device isn't connected. Dell already have TB3 ports on their laptops.
Personally, I won't buy a laptop without TB as it wouldn't be able to connect with my external display. A MacBook with TB3 rather than USB-C makes it a lot more versatile without changing the number of ports or design.
This is all fairly simplified and I'm nowhere near an expert on the topic - I'm sure someone can chime in on the other benefits of TB compared to USB and the like - but that's a rough overview.
