The above publication seems to be a revision or update of another USGS publication, USGS Bulletin 1532, Map Projections Used by the U.S. Geological Survey, by John P. Snyder, 1982, 313 pages. both seem to cover more or less the same material.

As every English schoolboy knows, a coordinate system can be laid out on the earth that provides coordinates in degrees that we call longitude and latitude. Latitude is our distance north or south from the equator. Latitude is zero at the equator, and +90 at the north pole. Longitude measures distance east or west from some arbitrary reference. This reference is the Prime Meridian, which has chosen to pass through the transit telescope at Greenwich Observatory in England. Most people denote (as I do) west longitude as negative, but by no means all.

A meridian is a line of constant longitude, passing through both the north and south poles.
A parallel is a line of constant latitude, parallel to the equator, and the parallels form a series of circles of every decreasing radius as you approach the poles.

The surface of the earth is nearly a spherical shape, but actually it is an ellipsoid, being flattened at the poles and bulging at the equator. The science of map projections is the study of various methods for displaying the curving surface of the eath on a flat (planar) surface. Any such effort involves some kind of compromise and choices to be made about what aspects of reality ought to be preserverd, and which can be distorted in some way.

A conformal projection preserves angles and angular relationships.
An equal area projection makes equal areas the same size at any point on the map.