An axis of revolution is a line around which something is revolved.
For example, consider a line segment on a plane. Imagine a line perpendicular to this plane, passing through one end of the line segment. If the line segment is rotated about this line, the resultant surface of revolution will be a circle with the radius equal to the length of the segment.
On the other hand, consider a line passing through one end of the segment which is not perpendicular to the plane of the line segment, but slanted, making an acute angle with the plane. Imagine now rotating the line segment around this line. Every point on the line segment will make a circle around the line, but the radii of each circle will be different, dependent on how far this point is from the line. The entire surface of revolution will be conical.
I see that ishpiro has done a great job explaining what an axis of revolution is. For my response, I thought it might be helpful to include a non-math example (I know sometimes these types of examples help me visualize concepts better).
Think of a merry-go-round: you have your plastic horses (or carriages, or whatever else people ride) going around that pole in the middle of the structure. In this case, that center pole is the axis of revolution because those horses are going around and around it.
That was a simple example, but there are some qualifications if you want to be more technically correct. An axis is a line and should therefore have no ends. Therefore, strictly speaking, you should imagine the pole as never ending, like it breaks through the top and bottom of the merry-go-round and extend endlessly.
Hope this helps!