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. The following image is from the Standalone version of Octane (figure 1).
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 (figure 2). 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