The "line" that you see when a cell divides is called a "cleavage furrow" or "contractile ring." It marks the position where new membranes will form to separate the two new cells--essentially the place where the two cells will pinch off from one another.

Pretend that a balloon is a cell (you can even go get one and blow it up -- but not all the way!) to test this. Now, take a string and wrap it completely around the middle of the balloon. If you tug on the two ends of the string, (like a drawstring on a duffle bag) you will see the balloon start to "cave in" along the same line as the string.

Now, think about a cell. The "string" is actually on the inside, just underneath the membrane and it is composed of actin and myosin (the same proteins that make up your muscles). This string of "contractile" proteins "pulls" (just like the drawstring) to
physically separate the two new daughter cells.

Here is a question for you: How does the contractile ring know where (and when!) to form and why is this important for normal mitosis?
Recall that during mitosis, the cell has to faithfully replicate all of the genetic material, then separate that genetic material equally into the two new daughter cells. What happens if the ring forms too soon? Not at all? What happens if the ring forms in the wrong place?

Scientists do not yet know exactly how the ring knows where and when to form, but it has something to do with the mitotic spindle (the cellular "machine" that pulls apart the chromosomess) and the stage of mitosis. If you are really interested in this and other kinds of cell biology, you can visit the website for the American Society for Cell Biology (www.ascb.org) and find out all kinds of cool things about cells and also about being a cell biologist.