How is the area of a country calculated?

I guess you could ask Google or WolframAlpha. Interestingly, these answers differ substantially. Perhaps, that's just a matter of how territories are interpreted but it illustrates the point that, there's really no easy answer. The question is at once terribly elementary and, on the other hand, fabulously interesting. Mandelbrot asked the question "How long is the coast of Britain?" Turns out that it depends strongly on how carefully you measure it.

So, the short answer is - it's super simple, in that you do it just like any other spherical polygon. Dealing with data at this level, as well as territorial disputes, is a bit more complicated. To illustrate more concretely, consider the image below.

You can see that, in a very simple sense, each "country" can be triangulated. These triangles can then be further sub-divided to get a good approximation and the areas of the triangles can be added up. Thus, again, on the most basic level it's no harder than adding up the areas of some triangles. To fully implement this, though, requires a fair understanding of computational geometry as well as access to solid data, which is likely disputable anyway.

The real question is - can you get good upper and lower bounds?

Addendum

Per request, here's a look at the other side of the planet:

And here's a look on the map of just India and its neighboring countries.

One easy way to do this would be a Monte Carlo method. Say you had a map of a country with map area $A_{map}$. Throw a large number $N$ of small seeds randomly on the map. By the number of seeds to fall within the country $n$, one can find the area of the country: $$A_{country} = A_{map} \cdot \frac{n}{N}$$ This of course ignores maps with projections that are nor area preserving, but never the less.

Geographers used to measure areas with a planimeter:

http://en.wikipedia.org/wiki/Planimeter

Nowadays I suppose they use something like ArcGIS, which has a built-in area calculator, but I don't know what algorithm is used.