This applet explores the Euler Line. You can verify the Orthocenter, Circumcenter and the Centroid. Then you can see how all three of those points are always linear.

OrthoCenter is the intersection of the three altitudues of the triangle. (An altitude is a line through a vertex and perpendicular to the opposite side) Verify this.
Use Control+Z to undo what you verified. (Cmd+Z on MAC)
Centroid is the intersection of the medians. (The median is the line from the vertex of a triangle to the middle of the opposite side) Verify this.
Use Control+Z to undo what you verified. (Cmd+Z on MAC)
Circumcenter is the intersection of the three perpendicular bisectors of the sides of the triangles. Verify this.
Use Control+Z to undo what you verified. (Cmd+Z on MAC)
Make a line (Euler Line) through two of the three points above (OrthoCenter, Centroid, Circumcenter). Move the verticies of the triangle and see how the Euler line moves as well.