Here is a simplified version of the PID formula: f(t)=Kₚ P(t)+Kᵢ I(t)+Kd D(t)
All we have done is simply take the full formula and replaced part of the terms with functions: P(t), I(t), and D(t).
The first component of the function, Kₚ P(t), is by far the most simple and easy to understand, as P(t) = e(t). For the sake of example, let's pretend that Kᵢ=0 and Kd=0 (a PID controller with only a proportional constant is known as a P controller). How will the system behave? Well, if the error is large, the output will be large. Likewise, if the error is small, the output will be small. Also, ideally, given enough time, the system always approaches its destination, assuming Kₚ is of the correct sign.
Say we apply this to a drivetrain. You want to drive a distance D, and you decide to set your motor powers using a P controller to accomplish this. In this case, your error is how far away the robot is from the desired location. As you start to drive forward, your error is large, so you drive forward quickly, which is desirable. After all, you aren't concerned with overshooting the target yet if you are far away from it.
But as the robot's distance to the target approaches 0, you will start to slow down, gaining more control over the robot. Once the error is zero, ideally, the robot will stop, and you have reached your destination. If you happen to overshoot, the error will become negative, and the robot will backtrack, repeating the process.