Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

As we can see from the figure, that two circles have two points in common. 

Two circles cannot intersect each other at more than two points. 

Let us assume that two circles cut each other at three points. 

But we already know that through three points only one circle can pass. 

So, two circles if intersect each other will intersect at maximum two points.