Given two points, there exist a unique line through them. There exist a unique perpendicular bisector of the line segment between those two points. Of course, any point on the perpendicular bisector is equal distance from the two points. That means that is you have three points, A, B, and C, say, take any two of the three segments defined by the three points, AB, and AC, say, and construct the perpendicular bisectors of the two segments, any point on one is equidistant from A and B and any point on the other is equidistant from B and C. The point where those two perpendicular bisectors intersect is equidistant from all three of A, B, and C. In other words, it is the center of the circle through the three points.
(What if those two perpendicular bisector do NOT intersect? In that case, they must be parallel which means that lines AB and AC are either parallel or are the same line. But since they have the point A in common they must in fact be the same line. We require that the three points NOT be on a single line to determine a circle.)
Two points are not sufficient because, given two points, we can choose any point on their perpendicular bisector as center and construct a circle through the two points. Since there are an infinite number of points on the perpendicular bisector, there are an infinite number of circles through any given two points.
If we have more than three points, choose three of them and construct the circle through them. Then choose a different three points and construct the circle through them. Those two circles are not necessarily the same- they might if all points happen to lie on a circle but in general they are not. Generally, there does NOT exist a single circle that passes through more than three given points.