Aspect
The aspect of a map projection is its orientation with respect to the earth's axis. Cylindrical and conic projections wrap around the earth, while planar projections touch it at a single point.

So for cylindrical and conic projections:

Wrapping around the equator gives you equatorial (normal) aspect.
Wrapping around the poles gives you transverse aspect.
Wrapping around the earth anywhere else gives you oblique aspect.

And likewise, for planar projections:

Touching the earth at any point on the equator gives you equatorial aspect.
Touching the earth on either pole gives you polar (normal) aspect.
Touching the earth at any other point gives you oblique aspect.

Perspective
A perspective projection is one that can be made geometrically, by projecting straight lines from a globe onto a developable surface. Perspective projections can be made onto cylinders, cones, and planes, but the planar ones are the most useful.

Different perspective projections result from moving the “light source” (the focal point of the straight lines) to different places. The rays can emanate from the center of the globe, or from a point on the equator (passing through the earth to the other side), or from an infinite distance.

Most map projections, however, are non-perspective mathematical constructions.

Classification
A handy way to classify map projections is by the appearance of their parallels and meridians.

Cylindrical	— straight parallels; straight meridians
Pseudocylindrical	— straight parallels; curved meridians
Conic	— partial concentric circles for parallels; straight meridians
Pseudoconic	— partial concentric circles for parallels; curved meridians
Planar	— concentric circles for parallels; straight meridians
Modified planar	— includes many variations on planar projections; no common appearance of parallels and meridians