Its all about how an application goes from "I would like to display X on a screen" to how X actually gets displayed. Wayland is effectively a language (technically a protocol) that graphical applications can speak to describe how they would like to be drawn. It's then up to a different program more deeply embedded in your OS to listen to and act on those instructions (this program is called a Wayland compositor). There's a lot more to it (handling keyboard input monitor settings, etc), but that's the general idea.
Wayland is a (relatively) new way of thinking about this process, that tries to take into account the wide variety of input and output devices that exist today, and also tries to mitigate some of the security risks that were inherent to previous approaches (before Wayland, it was very easy for one application to "look at" what was being displayed in a completely different app, or even to listen to what keys were being typed even when the app isn't focussed).
Thing is, change is hard, doubly so in the consensus driven world of Linux/FOSS. So, until the last couple of years or so, adoption of Wayland was quite slow. Now we're at the point where most things work at least as well in Wayland, but there's still odd bits of software that either haven't been ported, or that still rely on some features that don't exist in Wayland, often because of the aforementioned security risks.
I see X11 and Wayland as display protocols that tell to render things on the screen, for example to Desktop Environments like Gnome or KDE Plasma. X11 wasn't originally designed for this purpose, and its codebase is very messy and 'hacky,' which led to the development of Wayland.
DirectX is a set of APIs for game and multimedia development on Windows, whereas X11 and Wayland are display protocols that manage how graphical applications are rendered and interact with the desktop environment. DirectX is more similar to Vulkan in terms of providing a low-level API for high-performance graphics rendering or OpenGL.
it's a newer display server protocol designed to replace X11, focusing on improved security, simplicity, and contemporary display technologies such as multi-display variable refresh rate and HDR (eventually).
it's when devs of a graphics stack just suddenly feel the need to protect your own computer from itself, so they say fuck you to any features that they deem "insecure", including accessibility features (they will claim they fixed this, but it's opt-in per app. old apps will just be completely unusable for some people with special needs.)
But they eliminated tearing on the desktop! woo!!!!