Wikipedia has a good shortened version of the history of the islands and the various claims to sovereignty. I've cut and pasted the bulk of the history below.

There is controversy as to who was first to discover the Falkland Islands, with competing Portuguese, Spanish and British claims in the 16th century. The first reliable sighting is usually attributed to the Dutch explorer Sebald de Weert in 1600, who named the archipelago the Sebald Islands, a name they bore on Dutch maps into the 19th century.

In 1690, Captain John Strong of the Welfare en route to Puerto Deseado was driven off course and reached the Falkland Islands instead, landing at Bold Cove. Sailing between the two principal islands he called the passage "Falkland Channel" (now Falkland Sound), after Anthony Cary, 5th Viscount Falkland, who as Commissioner of the Admiralty had financed the expedition.

In 1764, French navigator and military commander Louis Antoine de Bougainville founded the first settlement on Berkeley Sound, in present-day Port Louis, East Falkland. In 1765, British captain John Byron explored and claimed Saunders Island on West Falkland, where he named the harbour Port Egmont and a settlement was constructed in 1766. Unaware of the French presence, Byron claimed the island group for King George III. Spain acquired the French colony in 1767 and placed it under a governor subordinate to the Buenos Aires colonial administration. In 1770, Spain attacked Port Egmont and expelled the British presence, bringing the two countries to the brink of war. War was avoided by a peace treaty and the British return to Port Egmont.

In 1774, economic pressures leading up to the American Revolutionary War forced Great Britain to withdraw from many overseas settlements. Upon withdrawal the British left behind a plaque asserting her continued claim. Spain maintained its governor until 1806 who, on his departure, left behind a plaque asserting Spanish claims. The remaining settlers were withdrawn by the United Provinces of the River Plate in 1811.

In 1820, storm damage forced the privateer Heroína to take shelter in the islands. Her captain David Jewett raised the flag of the United Provinces of the River Plate and read a proclamation claiming the islands. This became public knowledge in Buenos Aires nearly a year later following the publication of the proclamation in the Salem Gazette. After several abortive attempts, Luis Vernet established a settlement in 1828 after seeking authorisation from both British and Argentine authorities.

A dispute over fishing and hunting rights resulted in a raid by the US warship USS Lexington in 1831. The log of the Lexington reports only the destruction of arms and a powder store, though in his claim against the US Government for compensation (rejected by the US Government of President Cleveland in 1885) Vernet stated that the settlement was destroyed. The Islands were declared free from all government, the seven senior members of the settlement were arrested for piracy and taken to Montevideo, where they were released without charge.

In November 1832, Argentina sent Commander Mestivier as an interim commander to found a penal settlement, but he was killed in a mutiny after 4 days. The following January, British forces returned and requested the Argentine garrison leave. Don Pinedo, captain of the ARA Sarandi and senior officer present, protested but ultimately complied. Vernet's settlement continued, with the Irishman William Dickson tasked with raising the British flag for passing ships. Vernet's deputy, Matthew Brisbane, returned and was encouraged by the British to continue with the enterprise. The settlement continued until August 1833, when the leaders were killed in the so-called Gaucho murders. Subsequently, from 1834 the islands were governed as a naval station until 1840 when the British Government decided to establish a permanent colony