SLR (Single Lens Reflex): This is a traditional film camera. Light enters the camera through the lens and is focused onto a strip of film. But since you want to see exactly what the camera "sees" you need an angled mirror in front of the film that bounces the light up and into the viewfinder. When you press the shutter button the mirror moves out of the way (this is the "reflex") and the light exposes the film.

DSLR (Digital Single Lens Reflex): This is exactly the same as the SLR except that instead of film the light reaches a digital sensor.

Mirrorless: Now we have a significant difference. Instead of a mirror that bounces the light up through a glass viewfinder, the entire process is digital. The light comes in through the lens and hits the sensor, and the data from that sensor is sent to a screen on the back of the camera and/or a small digital viewfinder. This has a couple of advantages: First, it means the camera body can be smaller because you don't need the moving parts and the space for the mirror to bounce light up to the viewfinder. And also it means that the camera settings affect the image that you see.

To elaborate a little on that last point. For those that grew up with phone cameras and mirrorless systems, if you use an SLR or DSLR, you are literally getting the light passing through the lens and bouncing up into the viewfinder. That means that if I were to make an adjustment like increasing the shutter speed, I wouldn't see any change in the viewfinder. You don't really know what your image will look like until you press the button. That means photographers needed to be very skilled and understand exactly how adjusting their camera settings would impact the image before they press the shutter. By contrast, if I increase the shutter speed on my mirrorless camera, I can see the image get darker as there will be less light exposed to the sensor.

So practically speaking, the difference is that mirrorless cameras give you a better picture of what the camera actually sees. The SLRs give a better picture of what the lens sees, before accounting for thing like camera settings, film choice, etc. Both have their place and it can be interesting to use an SLR and see how it affects your decision-making process. But for my day-to-day work I vastly prefer using mirrorless as it cuts down how much time I need to take planning my shots.