Suppose 2 circles are constructed. under what conditions do the circles have 1 point in common, 2 points in common, no points in common, more than two points in common?

If the two circles are touching tangentially at one point then they have one point in common.  If centers are C1 & C2 and radii are R1 & R2. then  if distance C1C2 = R1+R2 then they have one point in common.

if 0< C1C2  < R1+R2  they are two points in common.
if C1C2 = 0  R1 = R2  then they have all the points in common. they are the same circle.