Start with just the axis and then position another line that is the same length, called the generatrix, parallel to it and distance r from it.
If you move the line so that it remains parallel to, and the same distance from the axis, it will trace out a circular cylinder. In the applet below, press Run to see the cylinder thus created.
The path traced out by one end of the line is called the directrix, and is usually a circle, but can be any curved path.
A way to think about this method is to consider the surface created as the locus of the generatrix. Just as the locus of a point is a line, so the locus of a line is a surface.
This way of defining a cylinder is in many textbooks and is presented here for completeness. However, it seems inadequate because it does not generate the bases of the cylinder. It defines instead a tube without ends. Normally a cylinder is defined as a closed surface which would by necessity include the two bases. This is also in conflict with the surface area of a cylinder which would usually include the area of the two bases.