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Globes give a good sense of the shape and size of countries but also limit your view to less than half the world at any one time.
If we unwrap a globe and lay it flat, the result is a projection that preserves neither the shape or relative size of countries.
15s: unfolded
We can improve this by stretching the projection, giving true land sizes but still distorting shapes.
25s: mercator
Mercator distorts everything and is, unfortunately, the projection used for Google Maps.
30s: mollweide
Mollweide is a vast improvement over Mercator for most uses. Like Lambert, it shows the true size of countries and continents at the cost of distorting shapes. But this distortion is minimized by projecting on to an oval rather than a rectangle.
50s: werner
Werner's projection is not in much use anymore but can be found in certain historical documents from the 15 hundreds. Also, you never know when you'll need a good heart-shaped map. Keep Werner in mind when wooing a carthographer.
65s: Collignon
Collignon might look strange but still manages to preserve relative sizes.
70s: Eckert
Eckert projections are some of my favorites. Each of Eckert's six projections have a different trade-off between the accuracy of shapes and sizes, and I find all of them to be visually pleasing.
85s: Fahey
Some people, not to be confused with flat-earthers, know that the world is round and therefore a round projection is needed.
90s: Foucaut
Other projections may appear of little practical use but they often have interesting mathematical properties that can be important, especially when comparing smaller map segments.
105: GitHub
This animation was created with Haskell and Blender. The full source code is available on GitHub.