The basis of the mercator projection is that of the globe projected onto a cylinder. The Conic projection is a globe projected onto a tangent cone. This means that meridians of longitute appear as straight lines and lines of latitude appear as parallel arcs. Wherever the projected 'cone' would touch the earth a standard parallel is drawn. In this projection the farther away from the standard parallel you get the more distortion occurs.
Conic projections are useful in portraying the middle latitudes in an east to west fashion.
The Lambert Equal Area projection assumes that there is a plane touching the earth at a single point. The closer to the center point you get the more accurate this type of map will be. Equal area means showing landmasses in their correct proportion. The Lambert Equal Area projection is used quite frequently for polar projections. The Miller Cylindrical projection is an adapted Mercator projection based on the suggestions of O. M. Miller.