As far as I know, the cylindrical robot type is one of the rarest nowadays. However, there are areas where they could still be useful. Also, they are always mentioned when stationary robot types are discussed, so I'll try to give you some insight on this topic.
In this picture you can see the basic structure of this type of robots. As you can see, it has three axes of movement - two of which are linear and one - circular. So, usually robots of this type can move along Z and Y axes and rotate along Z axis. As you can understand, this, basically, forms a cylindrical coordinate system - hence it has a cylindrical work envelope.
A little off topic. I have seen that SCARA robots sometimes are also regarded as cylindrical robots but this is not accurate. Although SCARA's work envelope also is cylinder-like, its structure and applications differs significantly.
Back to the topic. The most usual set of applications in which they are used is the type of applications where the cylindrical work envelope combined with horizontal tool orientation is required. For example - specific handling and assembly tasks or spot welding.
If I'm not mistaken, some kind of cylindrical robot was used in 1995 movie "Hackers". When Dade hacked into a TV station a robot changed VCR tapes. If I remember correctly, it could move along Z and Y axes and rotate around Z axis.
In this picture you can see a rather "ancient" robot. It's made somewhere in the now-nonexistent Soviet Union in 1985. It's a typical cylindrical coordinate robot manipulator that can move along two axes and rotate around the Z axis, thus positioning its horizontally oriented tool in a cylindrical coordinate system.
This is also an interesting example from another point of view. As you can see by the tubes, it uses pneumatics to move. Nowadays electrical power is dominant in robotics, so you won't see many robots like this anymore. My guess is that it was used in education.I hope this article too was valuable to you!