Kindergarten is the time for young children to develop the ability to interact effective with both peers and superiors. If your child has difficult being part of a group -- such as sitting in a group for story time -- he may need to repeat kindergarten. If he cannot deal with a mild frustration, such as having to stop playtime for naptime, first grade should perhaps be put off another year. Look also to your child's friendships: if he doesn't share well or respect personal space, repeating kindergarten could be best.
By the end of kindergarten, your child should be able to speak and pay attention to others for at least a few minutes. If she encounters difficult with letting others speak or has an extremely short attention span, she may need another year to develop. Kindergarten should also leave her able to form syntactically correct and coherent questions and answers and to understand basic directions. Your child should also be able to follow simple, two- or three-step instructions by the end of kindergarten.
Your child should repeat kindergarten if he doesn't know the alphabet, numbers up to at least 10 and most (if not all) letter sounds. If your child cannot yet read, he should still leave kindergarten able to decipher very simple three-letter words (such as "cat") by sounding them out. He should be able to write his name and a simple sentence (regardless of the spelling) before entering first grade. Check also that your child can pick out basic similarities and differences at the end of kindergarten.
An exceptionally small child may feel dwarfed and intimidated with a class full of normal-sized first graders and could likely benefit from another year to grow in kindergarten. Similarly, if your child started kindergarten by just barely making the age cut-off, she may not be mature enough to continue on to first grade. Giving her another year to develop emotionally can make a huge difference in both her performance and happiness at grade school.