The first map, above, is an equal-sized grid of the world, where 10 degrees lat/long is signified by a square (not a rectangle): http://www.legallandconverter.com/images/world.jpg . This is essentially a world map without a projection.
The Mercator Projection is the traditional projection used to turn the globe into a flat surface. It originates as a sea chart, i.e. a very practical use. More info: https://en.wikipedia.org/wiki/Mercator_projection and you can see the difference in country size below.
The Winkel Tripel Projection is seen by geographers as the best (fairest, most representative) projection. The National Geographic Society switched to this a few decades back. Shown below. More info: https://en.wikipedia.org/wiki/Winkel_tripel_projection
Robinson Projection also seen as good, but Winkel is generally preferred. This is what National Geographic used the Robinson for a long while, before switching to the Winkel Tripel. Info on Robinson here: https://en.wikipedia.org/wiki/Robinson_projection - image of Robinson projection is below, from http://alabamamaps.ua.edu/contemporarymaps/world/world/world3.jpg
Here's a good article summarising the pros and cons of the Mercator vs Peters projections, and other preferred projections: https://www.thoughtco.com/peters-projection-and-the-mercator-map-4068412