This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.

After doing this	Your work should look like this
We start with a triangle ABC.
1.  Find the bisector of one of the triangle sides. Any one will do. See Constructing the Perpendicular Bisector of a Line Segment.
2.  Repeat for the another side. Any one will do.
Optional step.  Repeat for the third side. This will convince you that the three bisectors do, in fact, intersect at a single point. But two are enough to find that point.
3.  The point where these two perpendiculars intersect is the triangle's circumcenter, the center of the circle we desire.   Note: This point may lie outside the triangle. This is normal.
4.  Place the compass point on the intersection of the perpendiculars and set the compass width to one of the points A,B or C. Draw a circle that will pass through all three.
5.  Done. The circle drawn is the triangle's circumcircle, the only circle that will pass through all three of its vertices.