I've decided to combine the HD-DVD and Blu-Ray pages into the same page due to the obscurity of the former and the short lived HD optical disc format war that occurred between 2006 - 2008. Unlike its rival, the HD-DVD is a discontinued optical media format and its entire history is pretty much linked with that of the still produced Blu-Ray. The success of the Blu-Ray is inherently tied to the demise of the HD-DVD during the beginning of its production.
It's pretty debatable as to why exactly the Blu-Ray won over the HD-DVD, as both formats support 1080p full HD video and both require a special blue laser to be played in a disc player, though there were some noticeable differences between them such as the fact that the HD-DVD could store 15 GB of data whereas a single sided Blu-Ray disc can hold up to 25 GB of data (for comparision, a standard single layer DVD can hold 4.7 GB of data).
However, it seems that the biggest reason as to why the Blu-Ray won over the HD-DVD is simply due to being promoted more than its competitor between having more major companies sign up to release movies onto the Blu-Ray to the mere fact that the Blu-Ray was patented by Sony, whom also released the PS3 the very same year as the release of the Blu-Ray and made it a standard for every PS3 to function as a Blu-Ray player (as they did with CDs for the PS1 and DVDs for the PS2). Microsoft's Xbox 360 did include an optional HD-DVD player at one point, though they had failed to standardize this in every model the way their competitor did and it was a completely optional add on.
While some companies did initially support the HD-DVD at first, they eventually gave up on the format as it was becoming clearer who the winner of this battle would be, and in 2008, the remaining companies who had initially supported the HD-DVD finally dropped their support over the format. However, although it may have won the battle to the HD-DVD, the Blu-Ray didn't seem to sell as well as expected and had to face off against stupid streaming services just a few short years later (you can guess who ended up winning here).
Today, the HD-DVD remains a niche format and only a small amount of physical media collectors own them and dedicated players for them.
You can learn more about the HD-DVD vs Blu-Ray war here:
While Blu-Rays are still in production today, they're really more like the modern day LaserDisc as DVD sales are still higher than that of Blu-Ray sales (except in Japan). This is likely due to the DVD generally being a more compatible format than the Blu-Ray and people already owning DVD players that can play DVDs and CDs, but not Blu-Rays, plus some movies and TV shows only having been released on the DVD but not the Blu-Ray. The general decline of physical media as people are increasingly willing to get themselves into subscription hell also seems to be taking its toll on both formats.