Get moving
Most cameras have a mechanical shutter consisting of two curtains. One that exposes the sensor and another that covers it (rear curtain).
If you go to a theater and see a show, the first thing you will see is also the last thing to see: the shoes of the actors (provided they are on stage!). That is not so smart in photography, as one end of the sensor would be more exposed than the other. By having two curtains – one that goes up like the curtain in the theater, and another that – in this metaphor – comes up the floor, you are certain all of the sensor gets the same amount of light.
The curtains move fast, but no matter how good technology you apply, it will take a bit of time to move the curtains. When we are talking shutter speeds at 1/8000th of a second for example, it takes – in relative terms – a long time to move the curtain. So to secure that the shutter is fast, the second curtain starts moving before the first curtain has revealed all of the sensor. And the faster the shutter speed, the closer to each other the two curtains will move. At very high speeds the curtains cover most of the sensor at any point of time and only a small crack of light is between them.
The max sync speed is the highest speed at which the camera at some point in time during the exposure reveals all of the sensor. Beyond that, the second curtain starts to move before the first curtain has revealed all of the sensor.
The max sync speed is a vital part of a cameras spec sheet if you want to use it for flash photography. When you go beyond the max sync speed, the flash needs to fire a series of flashes as the two curtains move across the sensor., also known has High Speed Synchronisation (HSS). If you shoot beyond the max sync speed without using HSS, you will get black bars or black areas in the image. The black areas are caused by the curtains on the move. HSS is very taxing on the flash and its battery as the flash has to fire several times in a very short while.
Related reading
What is flash duration? And should you care?
What is stroboscopic flash photography?
