The most common projections for planes are listed below.

Azimuthal Equal Area

Also known as the Lambert Azimuthal Equal-Area projection and presented by him in 1772 in his famous Beitrage collection.

While this projection is equal-area, created with this feature in mind, it is not a perspective projection. Used mainly on maps of the hemispheres and for world maps. The USGS has used this projection for a series of maps of the Pacific Ocean and National Geographic for its maps of the Moon.

Gnomonic

An ancient projection used by the Greeks. It is neither equal-area or conformal. All meridians are straight lines. All parallels are ellipses, parabolas or hyperbolas, except for the Equator and the poles.
The most useful feature of the Gnomonic projection is that it shows all great circle arcs as straight lines. It is not in common use, except for those needing to plot great circle routes.

                  This projection is supported in the spherical form only.

Azimuthal Equidistant

Not equal-area or conformal, it is another projection which has been in use for thousands of years. Presently, it is usually seen on insets of maps of the world showing the polar regions and hemispheres. In 1569, Mercator was one of the first to use it in this manner on his world maps when he demonstrated his now famous cylindrical projection. Maps which show the entire sphere are possible when using this projection although the distortion is extreme.
Aviators use Azimuthal Equidistant for their chart work, and it is especially useful for anyone wanting to track wave movement away from a central point. This is due to a unique quality of the Azimuthal Equidistant projection: distances radiating along straight lines from the center are true to linear scale. The position of any place along the line can be shown as a relative distance from the center.
Two variations of this projection are handled. In the first variation, the orientation of the positive Y axis is defined by a latitude and longitude of a point on the positive Y axis. In the second form, the azimuth of the Y axis is defined by an actual angle, east of north, of the Y axis. The specific variation to be used is indicated by the form variable. If an invalid form is provided, a coordinate system where the Y axis is true north is generated.

Azimuthal Equidistant with Elevated Ellipsoid

The Azimuthal Equidistant with Elevated Ellipsoid projection is a variation of the standard Azimuthal Equidistant projection that allows the user to specify ellipsoid elevation as a parameter.

Stereographic (Snyder)

Stereographic is conformal projection that is a true perspective projection, the only one that is both. It was possibly devised by Hipparchus in the second century BC. The observers point of view is just above the surface of the earth.
The Snyder variation of the Stereographic is used mostly in the U.S.A. For other regions of the world, it is more common to use the other variety, "Oblique Stereographic."

Oblique Stereographic

Used most commonly for stereographic mapping regions outside the U.S.A.

Polar Stereographic

Used most often for maps of the polar regions by setting the Standard Latitude to 90 degrees. This method can be used to map the polar regions of some of the planets in our solar system.

The central meridian is a straight line. All other meridians and parallels are arcs of circles.
For more information see Stereographic (Snyder).

Polar Stereographic with Standard Latitude

Identical to Polar Stereographic Projection except that it uses a user-specified standard latitude instead of scale reduction factor to determine point/circle which is tangent/secant to the mapping plane.

Orthographic

This projection puts the viewer somewhere in space, looking back at the Earth. It has the appearance of a globe and only one hemisphere can be seen at one time. It is an ancient projection, in use by the Greeks and Egyptians thousands of years ago.
It is not equal-area or conformal. Meridians and parallels can be straight lines, circles or ellipses with large distortion around the edges.
Two variations of this projection are handled. In the first variation, the orientation of the positive Y axis is defined by a latitude and longitude of a point on the positive Y axis. In the second form, the azimuth of the Y axis is defined by an actual angle, east of north, of the Y axis. The most common case is a coordinate system where the Y axis is true north.