To begin, children should understand what a circle is. You can explain to much younger children that a circle can be considered a line that has simply curved around to touch at both ends. You can illustrate this best by cutting a piece of string for each child about 1 foot long, and then showing them at a table how to move the string and form a circle by joining the two ends.
Smaller children probably won't understand what a word like "circumference" means -- the distance around a circle -- but it's easy to show them what it is. Have a large group of children form a circle, and then instruct another child to walk around the circle, counting the number of steps it takes to complete a full turn. The total number of steps is a loose interpretation of a circumference. Another, more accurate way to show the method is to give children several cans of food, in different sizes, along with some string, and then show them how to wrap the string around the can one time. Use some small scissors to cut the excess string from each child's can, and then use a ruler for the string that wrapped the can. The resulting measurement is an accurate reading of the can's circumference.
For older children who can do basic multiplication, circumference is a little easier to explain. Instruct children to draw a large circle on a piece of paper, using a mathematical compass then, using a ruler, draw a line through the center from one side to the other. After they've done this, have them measure the line they drew. This is the diameter of the circle. Once this measurement is done, the fun part comes: teaching about pi, which is a constant, non-exact integer. You don't have to go all-out in explaining the billions of possible integers it can represent, but you can talk about how it was created and why. For intensive purposes, have the constant pi equal 3.14. Multiplying this integer by the diameter of the circle gives its circumference.
Another way to get the circumference of a circle is to show how to measure the radius of a circle, which is the distance from the center of a circle to one end. Older children again can be taught this by instructing them to draw another circle on paper, and then using a thumbtack, poke a hole through the approximate center of it. Once done, show them how to measure from the hole to the edge of the circle with a ruler. Now that they know the radius, present the circumference formula as: pi x r (radius) x 2. Then, help the children calculate the result. Once they have the answers, add some fun to the instruction by having them measure the diameter of the circle again, and multiply it by pi, or 3.14 to see how accurately they judged the center of their circles.