Once upon a time there were analogue connections, where signals varied continuously, and were basically used to directly control the wavy bits of metal inside TVs and monitors that beamed electrodes at the phosphor at the front of the screen. And this was fine.
Then there was the move to LCD screens. Which came first to PCs, because monitors are much smaller than TVs, and people were happy to spend a fortune on getting nice thin ones on their desks. And it seemed silly to change a digital signal inside the computer into an analogue one, sending it to the monitor, and having it convert that signal back into a digital one that turned different pixels on and off. And so DVI was born - the Digital Video Interface. And that allowed the digital signal to go direct from the computer to the monitor, and all was good.
And then the TV people decided they'd also like to get in on the digital action, now that LCD TVs were selling. But they didn't like the idea that people could pull the awesome digital signal direct from the cable and use it to produce pirate copies, so they wanted a connector that included encryption*. They also wanted sound included, which the PC people hadn't cared about, because monitors don't tend to have speakers in them. And rather than building on DVI, they decided to produce HDMI instead, which did these two extra things.
And so we have two competing cable types in the digital world: DVI for PCs and HDMI for TVs. Except that a lot of more recent monitors will also take an HDMI connection and a fair number of decent TVs will take a DVI connection.
I suspect that in the long run HDMI will win, as they're constantly upgrading the spec to allow for higher resolutions**, but it's probably going to take a while.
*This was, of course, completely successful. As you can tell by there being no pirated videos anywhere on the internet. At all.
**Up to 3840x2400 now. Which is twice as high and twice as wide as Blu-Ray currently does, and about the same number of pixels as they use for cinemas. Unless your TV is ridiculously big, you don't need to go that high.
Original post on Dreamwidth - there are comments there.