We need to define what we mean by "stop-start" and "constant-run":
Stop-Start: This means that a pressure switch is used to automatically turn on and off the compressor at designated low and high-pressure levels. When switched on, the pressure in the tank builds to a certain psi (pounds per square-inch) and then automatically turns off. When air is used, the psi in the tank is reduced and eventually reaches a low setting where the pressure switch automatically turns back on to build psi back to the high point. This operation is common for smaller electric, direct-drive compressors and some smaller wheelbarrow electric compressors.
Constant-Run: This means that the motor continues to run and as a result continues to turn the pump. When the tank reaches the high-pressure setting, a pilot unloader-valve opens to atmosphere so that the compressor can continue to run. The valve will close again when the psi is at the low point and begin to direct air back into the tank and re-build to the high-pressure point. This setup is used more often when using tools that require a heavy and constant flow of air without interruption, such as a grinder or sander or sprayer.
(NOTE: Normally if a compressor stops and starts more than 20 times per hour, you need to use a constant-run setup. If not, you are in danger of ruining the motor due to excessive heat buildup. Stopping and starting so often creates a huge amount of heat, more than if the motor stays running all the time.)
You can find compressors that have a "dual-control" option built on to the compressor where you can switch back and forth between stop-start and constant-run depending on the situation.
All gas-powered compressors are set up with this unloader-valve system because an engine cannot stop itself and start up again. It will only idle-down and speed-up at set psi levels via a throttle-control mechanism.