The reference (in the US, at least) is John Snyder's Map Projections--A Working Manual. The entire monograph is available as a Google book.
Introductory sections give the theory. The theory is accessible to someone with a working knowledge of multivariate calculus. Emphasis is on documenting formulas, primarily series expansions needed for subsequent calculations. Detailed derivations of most formulas are not worked out. (Snyder was not a mathematician and became interested in projections only later in life. His emphasis--given this was written decades ago when one was lucky to have a Fortran compiler available and a few seconds of CPU time on the local mainframe--is on documenting formulas that could be converted to working code.)
The bulk of the book is devoted to describing 26 major projections organized by type: cylindrical, conic, azimuthal, "space map," pseudocylindrical, and miscellaneous.
Each description begins with a bulleted summary of properties and then about a page of historical information. Following this are a narrative of the features of the projection--including a detailed line drawing with a lat-lon graticule--formulas for the sphere (projection and unprojection), and formulas for the ellipsoid.
Appendices include extensive numerical examples of the calculations (108 pages!) and some US-specific information about USGS projections and the State Plane coordinate system.