Mercator, though a map-projection by that term's general meaning, isn't made by actual projection.
There is a map made by projection onto a cylinder: Gall Stereographic. A cylinder is tangent to a globe around the globe's equator. For a particular point on a particular meridian, a line is projected from the equator directly opposite that meridian, through that point on that meridian, and that line is extended to the cylinder. Where it intersects the cylinder, that's where that point on the globe is mapped on the cylinder. So the projection is from the point on the equator opposite the meridian on which lies the globe-point to be projected onto the cylinder.
Then the cylinder is expanded north-south so that shapes will be perfect along the 45th parallel.
Gall-Stereographic was one of 3 cylindrical projections proposed by James Gall, a Scottish minister, in 1855.
He also proposed an equal-area projection, called Gall-Orthographic, at the same comference. It was later reinvented by Arno Peters, and known as the Peters Projection, or the Gall-Peters Projection.
It too was constructed by an actual projection of a globe onto a cylinder. In this case each point on the globe is projected onto the equator-tangent cylinder by a horizontal line (the globe's axis is vertical), originating at the globe's axis, extending out horizontally, through a point on the globe, and out to the cylinder. Where it intersects the cylinder is where that globe-point is mapped on the cylinder.
That's Lambert's Cylindrical Equal-Area. Gall expended it north-south to give perfect shapes at latitude 45 (just as he did with his Stereographic projection).
Gall Stereographic is very popular as a world wall-map. It was Gall's favorite. Gall Orthographic, better known as the Peters Projection, is also quite popular with some school-districts, and many humanitarian organizations.