Yellowstone is so unique because of the mixture of magma and water, from a tectonic "hot spot" that's been in this mantle location for around 20 million years, a the water that is naturally in the crust at this location.
Everyone kinda understands Plate Tectonics, yeah? The idea is that the top crust of the planet is made up of rigid plates that are sort of floating on the upper mantle, so they move around, like a Slip n Slide. This movement causes most of the geological activity on the planet, like earthquakes and volcanic eruptions.
Most, but not all.
Mantle activity known as "hot spots" are thought to be the result of plates being subducted under each other millions of years ago. See, when one plate is forced under another, it starts to loose water as the water in the crust heats up, being surrounded by molten magma. That water then "irritates" the mantle, to say it simply. And even once the plate has long been completed subducted and melted away, the irritation remains, causing the hot spot.
So a new plate moves over this area of the mantle, and magma wells up from the old irritation. Essentially, hot spots are thought to be evidence of much, much older plate movement. Currently, the hot spot is over a good chunk of Wyoming (remember, the hot spot doesn't move, the plate does!).
There are lots of other hot spots around the globe, Iceland being one of the more famous. But the crust in Northern Wyoming is very different than the crust over other hot spots, because it's got so much water inside it. There's an entire network of caves and underground lakes in this region, as well as a lot of natural gas. That's the reason for the numerous geysers and other thermal features in Yellowstone National Park and the surrounding area.