The simplest lens is a single lens - like a magnifying glass. This would give a lot of aberrations, and quickly became unpopular among photographers.
The next step was to make the lens curved, with the concave side facing the subject. This got rid of a lot of the worst aberrations, but still not good enough.
Then came the "anastigmat": Two lens elements, one positive, one negative, cemented (usually) together. This was a lot better, and actually usable for photography. There was still a bit of chromatic aberration, though... Removing the front group of a Symmar or similar will show you what these are like: Not at all bad, but far from perfect...
The "double anastigmat" is, with a few variations, the basis for all modern lenses. This consists of two anastigmats with a bit of space (and an aperture) between. The convex sides of both groups face out (from the aperture).
The Tessar is a double anastigmat with the rear anastigmat uncemented - 4 elements in 3 groups.
The Heliar (original) is a double anastigmat with an added negative element midway between the main groups.
A variant of the double anastigmat is the "doppelte Gauss", or double Gauss lens. This uses a more extreme curvature to the elements, as well as slightly different glasses, to give wide field coverage. Angulos, Super Angulon, Wide field Ektar, etc. are all variants of this.
Adding even more elements can refine the properties of the lens, or change them altogether. Very few LF lenses have more than 6 elements, or more than 4 groups. Very few lenses for 35mm have fewer than 6 elements or 4 groups...