This is the stepbystep, printable version. If you PRINT this page, any ads will not be printed.
See also the animated version.
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 compasses' point on the intersection of the perpendiculars and set the compasses' 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. 