Cylindrical projection wraps texture maps on a surface with a cylindrical shape. Cylindrical projections provide a quick way to map a texture on cylindrical-shaped surfaces without too much distortion. However, the seams of the texture may be visible in the render, depending on the shape of the surface.
Figure 1: The result of using Cylindrical projection with the Checks texture node
This projection performs cylindrical mapping where the U coordinate is the longitude, and the Y coordinate is the world space Y coordinate. For Images, the mapping on the Y axis maps the image to the [-1, 1] interval. For Procedural textures, the W coordinate is the distance from the Y axis. For points on the ground plane (Y = 0), cylindrical and spherical mappings now map to the same points on the images, or what would be the equator on spherical mapping.
Figure 2: Cylindrical projections on a box, cylinder, and sphere