This topic is based almost verbatim on the text in An Album of Map Projections, U.S. Geological Survey Professional Paper 1453, by John P. Snyder. It is included so Manifold users can benefit from the words of the master himself.
The advent of computerassisted cartography has now made it much easier to prepare maps. A map can now be centered anywhere on the globe, can be drawn according to any one of many projection formulas, and can use cartographic data files having a level of detail appropriate to the scale of the map.
An equalarea map projection correctly represents areas of the sphere on the map. If a coin is placed on any area of such a map, it will cover as much of the area of the surface of the sphere as it would if it were placed elsewhere on the map. When this type of projection is used for smallscale maps showing larger regions, the distortion of angles and shapes increases as the distance of an area from the projection origin increases.
An equidistant map projection is possible only in a limited sense. That is, distances can be shown at the nominal map scale along a line from only one or two points to any other point on the map. The focal points usually are at the map center or some central location. The term is also often used to describe maps on which the scale is shown correctly along all meridians.
An azimuthal map likewise is limited in the sense that it can correctly show directions or angles to all other points on the map only with respect to one (or rarely two) central point(s).
A conformal map is technically defined as a map on which all angles at infinitely small locations are correctly depicted. A conformal projection increasingly distorts areas away from the map’s center point or lines of true scale and it increasingly distorts shapes as the region becomes larger but distorts the shapes of moderately small areas only slightly.
Consistent with these definitions, maps simultaneously exhibiting several of these properties can be devised:

The above table tells us that a conformal map can also be azimuthal, but it will not be equal area and it will not be equidistant. An equal area map cannot be conformal and it cannot be equidistant but it can also be azimuthal.
A map projection may have none of these general properties and still be satisfactory. For example, the Robinson projection is well suited for thematic presentations involving the entire Earth, but it has none of the above properties. It looks good but it is not conformal, not equal area, not equidistant and not azimuthal.
A map projection possessing one of these properties may nevertheless be a poor choice. As an example, the Mercator projection continues to be used inappropriately for worldwide thematic data. The Mercator map projection is conformal and has a valid use in navigation but very seriously distorts areas near the poles, which it cannot even show. It should not be used (although it frequently is) for depicting general information or any arearelated subjects.
From the perspective of design as well as distortion reduction, a projection may be selected because of the characteristic curves formed by the meridians and parallels. Using nomenclature suggested by L.P. Lee, we use the following terms to describe projections:
Cylindric: Projections in which the meridians are represented by a system of equidistant parallel straight lines, and the parallels by a system of parallel straight lines at right angles to the meridians.
Pseudocylindric: Projections in which the parallels are represented by a system of parallel straight lines and the meridians by concurrent curves.
Conic: Projections in which the meridians are represented by concentric circular arcs and the meridians by concurrent curves.
Pseudoconic: Projections in which the parallels are represented by concentric circular arcs, and the meridians by concurrent curves.
Polyconic: Projections in which the parallels are represented by a system of nonconcentric circular arcs with their centers lying on the straight line representing the central meridian.
Azimuthal: Projections in which the meridians are represented by a system of concurrent straight lines inclined to each other at their true difference of longitude, and the parallels by a system of concentric circles with their common center at the point of concurrency of the meridians.
Note: In a few cases the projection is geometric but in most cases the projection is mathematical to achieve certain features.
Regular Cylindrical  Specifying latitude and longitude origin centers the projection about a given location. Various cylindrical projections allow the specification of standard parallels that are used to govern the mathematical mapping from sphere to cylinder.
Transverse Cylindrical  Specifying latitude and longitude origin (or in some cases, the standard meridian) centers the projection about a given location. Various transverse cylindrical projections allow the specification of standard parallels that are used to govern the mathematical mapping from sphere to cylinder
Oblique Cylindrical  Specifying latitude and longitude origin (or in some cases, the standard meridian) centers the projection about a given location.
Regular Conic  Illustrated with the latitude and longitude origin at the pole. Specifying a lat/lon origin other than the pole results in an oblique projection. Standard parallels, if specified, define the mathematical mapping from ellipsoid to the cone.
Polar Azimuthal (plane)  The projection’s latitude and longitude origin is at the pole.
Oblique Azimuthal (plane) The projection is centered at a latitude and longitude origin other than the pole.
Three traditional rules for choosing a map projection were at one time recommended as follows:
For lowlatitude areas: cylindrical.
For middlelatitude areas: conical.
For Polar Regions: azimuthal.
Inherent in these guidelines was the idea that it would be difficult to recenter a map so that the area of main interest was near the area of the map that has the least areal or angular distortion. On the other hand, with Manifold it is easy to recenter the projection to any location.
As a result, a projection no longer needs to be rejected merely because previous uses were traditionally centered inappropriately for the desired application.
In fact, the mathematical form of many projections (as implemented in Manifold) permit the user to alter the form of the map to reduce the distortions within a certain area. Most commonly, such alteration is accomplished by establishing standard lines along which distortion is absent; often, these lines are parallels of latitude. The Albers EqualArea Conic projection, for example, will often be used with two standard parallels to "customize" the projection to a particular region. However, most properties of the map projection are affected when a standard line is changed. One should therefore keep track of what parameters were used in a particular projection.
Another way of altering the relationships on a map is by using different aspects, which involves moving the center of the projection from the normal position at a pole or along the Equator to some other position. As with the case of custom standard lines, one should keep track of any recentering done with a given projection.
When a map requires a general property, the choice of a projection becomes limited. For example, because conformal projections correctly show angles at every location, they are advisable for maps displaying the flow of oceanic or atmospheric currents. The risk of using a conformal projection for a worldwide map is that the distortion of areas greatly enlarges the outer boundaries, and a phenomenon may seem to take on an importance that the mapmaker did not intend. Equalarea maps should be considered for displaying arearelated subjects or themes, such as cropgrowing regions.
Once the purpose of a map has been decided, the geographical area to be included on the map must be determined. This may be a region, or the entire world. The larger the area covered, the greater is the Earth’s curvature involved in the map. If the map area is a region, then its shape, size and location are important determinants in making decisions concerning projections.
Bearing in mind these determinants, mapmakers can apply traditional rules of choice, such as those mentioned above, or they can study the patterns of distortion associated with particular projections. For example, azimuthal projections have a circular pattern for lines of constant distortion characteristics, centered on the map origin or projection center. Thus, if an area is approximately circular and if its center is made the origin of the projection, it is possible to create a map that minimizes distortion for that map area. Ideally, the general shape of a geographic region should be matched with the distortion pattern of a specific projection.
An appropriate map projection can be selected on the basis of these principles and classifications. Although there may be no absolutely correct choice, it is clearly possible to make a bad judgement.
Mapping experts will choose from a wide array of projections to meet specific objectives as discussed above. For general use, most people will usually develop a favorite projection for a particular situation and then use that projection over and over. For most situations either Orthographic or Lambert Conformal Conic are good choices.
The following projections work well for most users:
Region to be mapped 
Suggested Projections 
Whole Earth 
Robinson (pseudocylindrical) or Miller Cylindrical . Robinson seems to be fashionable for thematic maps. Any of the pseudocylindrical projections will be fine if you like their appearance better. 
Hemispheres 
Orthographic (azimuthal) for a "view from space" look, and Lambert Azimuthal Equal Area for thematic maps where the relative size of countries near the edge of the projection is to be preserved. 
Continents 
Use Lambert Conformal Conic for North America and Eurasia. Use Lambert Azimuthal Equal Area or Orthographic for South America and Africa. Use Orthographic for Australia, and Antarctica. 
EW Countries or Regions 
Use Lambert Conformal Conic for US, Canada, Russia, and China. Use either Lambert Conformal Conic or Orthographic for Europe. Use Orthographic or Lambert Azimuthal Equal Area otherwise. 
Polar Regions 
Orthographic or Lambert Azimuthal Equal Area. 
Oceans 
Orthographic or Lambert Azimuthal Equal Area.

Smaller Countries or Regions 
Orthographic or Lambert Azimuthal Equal Area. 
NS Countries, Oblique Regions 
Long, thin countries aligned NorthSouth such as Chile are one of the few times we would use Transverse Mercator. Oblique regions like the Alaska panhandle are mercifully rare: Use the Oblique Mercator in such cases. 
Other than personal taste in visual appearance there are two primary reasons not to use one of the above projections:
A specific projection is required to match other data available or to exchange data with other users.
The technical characteristics of a particular projection such as preservation of scale or relative area are required for creating a paper map or other presentation.
Manifold provides a very wide array of projections in addition to the standard projections mentioned above.
If Orthographic or Lambert Conformal Conic work well for almost all mapping that does not involve the entire world, why are systems like Universal Transverse Mercator (UTM) so frequently encountered? These systems came into use in a day when any mapping or projection computation had to be painfully prepared by hand. They were developed to solve problems within the technological limits of their day. They live on in modern times as living fossils simply because so many maps have been prepared using them that their usage has gained momentum. In some cases, the use of projections like UTM is required by law.
Modern desktop computers make it possible to compute a precise, perfectly centered projection for any location on Earth so in modern times we can use whatever projection works best for all the factors we wish to consider. We can completely ignore the cost of projection since there isn't any cost. We can therefore choose projections that minimize distortion such as the Lambert Conformal Conic or provide a very natural look like the Orthographic, and compute each projection to fit perfectly the view desired.