This is probably a little weird to anybody new to PC gaming, but back in the 90's there were often multiple graphics drivers to choose from, each offering radically different hardware support and performance. You had proprietary drivers that would only work with certain cards, like 3DFX's glide, and you had open ones that worked with many cards.
Direct3D, which was the 3D API component of DirectX, essentially took over the entire market and has been dominant for over a decade now.
When you are talking about Vulkan and DirectX and OpenGL, you are basically talking about the middle-layer that lets your software talk to your hardware to get everything working. Today's GPUs are much more than merely tools to handle graphics and can handle many different kinds of tasks. They are essentially miniature computers well suited for all sorts of multimedia and parallel performing functions.
The nature of these middle layers has far reaching effects for computing beyond video games. A major reason why software that really uses these GPUs is seemingly always limited to windows is because Direct X only runs on windows. And recently, only specific versions of Direct X runs on specific versions of windows. You can have a PC whose hardware is perfectly capable of doing advanced things, which cannot access these advanced functions because the right direct X middle layer is not available on your version of windows. Vulkan aims to be available as many places as possible, even beyond windows.
