Spain attacked the British colony bringing the two countries on the brink of war, which was settled by Spain capitulating and letting the British settlement be re-established. However, Britain abandoned the Falklands in 1776 leaving it all to Spain (although also leaving a plaque asserting British sovereignty). Spain similarly abandoned the islands in 1811, also leaving a plaque. From this point on for some years, the Falklands ended up being under no effective control, but being a harbour for various fishing, whaling and other vessels.
Argentina gained independence in 1816, and in 1820 had sailed to the islands and asserted sovereignty over them. Between then and 1833, merchant Luis Vernet sought permission to settle there from both the Argentines and the British. He received assurances, and the Argentines appointed him Governor in 1829, to British (and US) protest. However, by 1833 Britain had re-established itself on the islands and ordered the Argentines to leave, which they did. A British colony was established and has remained relatively undisturbed, notwithstanding the more recent Falklands War.
Argentina claims it was first, as it inherited the French then Spanish settlements, and was forcibly ejected from the Falklands. The UK claims that it has a parallel claim, that the Spanish abandoned the Falklands (like the British did), and there was no indigenous or Spanish/Argentine colony established before the British colony. Moreover most of the current population opposes Argentine sovereignty.
In essence, for all of the debate the population of the Falklands do not want to be governed from Buenos Aires. Arising from this are claims to exploit the Exclusive Economic Zone around the Falklands for energy exploration. The beleagured Argentine government is seeking to distract attention from its own economic mismanagement by confronting the UK over this.
The Obama Administration's response? Neutrality.
According to the Times ""The Obama Administration “is trying to split the difference as much as it can because it knows that coming round to the British position would again create a lot of ill will in the region"
The leftwing Argentine government, beleagured by high inflation, is challenging British attempts to take advantage of the UK EEZ as it surrounds the Falklands.
Argentina's claim will no doubt have the backing of the cabal of socialists that now run many Latin American countries, none of whom give a damn that most Falkland Islanders want to remain British.
The bigger point is that the "special relationship" is over. The Obama Administration is reverting to the form REJECTED by Ronald Reagan, the realpolitik preferred by the State Department.
Who will know if John McCain would have done the same, would George Bush have just thumbed his nose at Britain given its close support in Afghanistan and Iraq?
One thing IS sure, from the DVD set gifted to Gordon Brown to this, there can be no question that the Obama Administration doesn't think the UK deserves consideration beyond that of just another friendly country - like Argentina, France or South Africa.