As it turns out, I don't think these alternate names are encountered all that much, but they can certainly help depending on what you're reading. I've come across some of these in newspapers and online publications.
By this point you may be wondering what I'm talking about exactly. Well, as you probably know, the Chinese language uses kanji exclusively. And some of the Chinese names for countries were carried over into Japanese. Over time, however, these Chinese names have come to often be abbreviated in Japanese; reduced to one or two characters to indicate a particular nation or culture. Here's a partial list (link from Tofugu).
As Hashi notes in his piece at Tofugu, these days country names are most often written in katakana. However, there are cases where the kanji are used, especially in 熟語 (じゅくご, kanji compounds).
For those who
The United States - 米国 (べいこく) - literally "rice country."
France - 仏国 (ふっこく; though my Japanese input won't even recognize that to convert to kanji) - literally Buddhist or "saint-like" country.
Italy - 伊 - (い) - although I suspect this must have some other meaning, according to my searches, this kanji is used pretty exclusively to mean "Italty"
Germany - 独国 (どくこく) - according to Wikipedia, 独 was converted from 徳 in Chinese. Literally Germany was/is called "virtuous country."
The Netherlands - 蘭 (らん) - "orchid."
Although it seems to often take some research to find out what the original intentions behind some of these names were (if it is even known), it's pretty cool to see how some of these countries are named. "Virtuous country" and "rice country" for example. Of course in many cases these are just cut down versions and abbreviations of longer names, so perhaps they are just nonsense at this point...
I think if you notice these, they will often be in compounds. One that is used quite often in newspapers is 日米関係 (にちべいかんけい), which means "Japan-America relations."
For those of you familiar with these, any favorites or others you'd like to share?