It was a very short war (10 weeks) between Argentina and the United Kingdom for the control of the Falklands Islands, in the South Atlantic.
But the most important fact is: it was the last war in the history where two western countries fought on opposite sides.
I doubt Thatcher considered Argentina Western..
Whether she considered it western or not is irrelevant. Argentina is in the Western hemisphere, speaks a European language, is overwhelmingly Christian, a highly developed nation, and, at the time, was firmly pro-US. Heck, if you want to bring "race" into it, Argentina is even pretty "white", and the vast majority of its population is of European descent. There is no way that it's not counted as Western.
But anyways, the war was a blatantly imperialist move by Argentina, ironically enough. The Falklands are British.
The Western world is a very vague concept with very different definitions applied in different parts of this "civilization". In Northern Europe we tend to view it as non-Orthodox Europe + the four Anglophone White majority countries outside Europe.
Europeans and Latin Americans have a very ambivalent relationship and non-Hispanic Euros and Latin Americans do not really consider each other as belonging to the same category. So a war between a European and a Latin American country is not more exotic than a war between a sub-Saharan African country (with European colonial heritage, European official language and (often) predominantly Christian) and a European country.
North Americans are of course somewhere in the middle of this and will tend to view the two groups as both related to you and therefore part of a common family of nations.