dK's summary above is spot on.
What is a Pitcher’s Circle?
The pitcher’s circle is the chalked circle around the pitcher’s rubber, also commonly referred to as “plate”. The circle has a diameter of 16 feet. This means that the circle observes an 8-foot radius which starts right from the middle of the front surface of the rubber.
So when does the pitcher’s circle come into play?
The play is said to start when the pitcher carries the ball inside the circle. And with the foot on the plate, she then releases the ball.
Once the ball is thrown back by the catcher or fielder to the pitcher, provided that the pitcher is now in the circle and has full control of the ball, the play is then said to be “dead.” In such case, the batter reaching the first base must have her foot on the base until the pitcher releases the ball. Additionally, each runner standing on the other bases will have to stay put as well.
Hence, the pitcher’s circle is basically used to establish that the play is complete or has ended. Again, this means that the pitcher is now inside the circle and in control of the ball.
Another purpose of the pitcher’s circle is to serve as a basis for the look back rule.
Look Back Rule Defined
The look back rule stipulates that when the pitcher is in control of the ball inside the pitcher’s circle and provided that she is not threatening to make a throw or fake it, the following must be observed:
1. Any runner standing on a base must remain on the base.
2. Any runner who is not on the base must either go back to the previous base or must advance towards the next one. Either action should be done without any hesitation. Otherwise, any attempt made by the runner to change direction or stop or dance around, can be a valid ground for her to be called out.
The look back rule is actually implemented in order to cut to the chase and prevent the pitcher and runner from unnecessary playing “cat and mouse” with each other.
What happens when the feet are on the line of the circle?
When both feet are on the line of the circle, then the pitcher is still considered to be inside the pitcher’s circle. As a rule of thumb, the player is considered to be inside the circle for as long as any part of the foot is on the line of the circle. This practice also holds true for any other line on the field.