This map projection is an equal-area map projection which displays the world on an ellipse.
However, it is completely unlike the Mollweide projection. In the conventional case, the parallels are curved, and there is no stretching at the center of the map.
This map was created from a hemisphere of the equatorial case of Lambert's Azimuthal Equal-Area projection using a trick similar to the method by which the Miller Cylindrical projection was made from the Mercator projection.
First, place the whole world in one hemisphere by the simple expedient of dividing all longitudes by two, then to compensate for that, stretch the map out twice as wide.
Note that Adams' Equal-Area projection, which spaces the parallels as they are spaced on the central meridian of an equatorial Lambert's Azimuthal projection, could be thought of as the limiting case of this kind of manipulation.
This projection is also popular in the oblique case, producing maps like this:
Compare the Mollweide projection in precisely the same orientation:
This projection was devised by Hammer, using the idea first used by Aitoff in making a compromise projection in such an ellipse from the Azimuthal Equidistant projection. Aitoff, however, then returned the compliment by using the basic method behind the Lagrange Conformal projection, but this time superimposing one equal-area cylindrical projection upon another. The ratio in scales was 9 to 10, and this time a compensating stretch was needed, to create an equal-area projection which I find quite appealing:
However, to make the pole lines shorter, to try to make the result a more balanced compromise between the Hammer-Aitoff on the one hand, and the cylindrical equal-area on the other, I used the sine of 75 degrees, or .9659258, as the factor to produce this projection:
However, that didn't seem to be an improvement, since the extra shearing seems to be worse than the stretching in the previous projection. However, in an oblique case of the projection:
it's possible to move all the important land masses out of the areas with high shearing, producing quite a pleasing result.