Bisecting a line segment
Open a compass to more than half the length of the line. Place the compass's point on one endpoint of the line, and draw a circle centered at it. Repeat for the other endpoint of the line. Draw a second line connecting the intersections of the circles. Where the second line meets the first is the midway point for both.

Bisecting an angle
At the vertex of the angle, place the point of the compass. Draw an arc with the compass that strikes both the angle's legs. Where the arc meets each leg, place the compass point there and draw a circle (leave the compass at the same radius as before). The circles intersect at two points: one is the vertex of the angle, and if you draw a line connecting it to the other point, you'll have cut the original angle straight in half.