Argentina has been laying claim to the Malvina Islands for many years and for many reasons, primarily patriotic ones.
Having a set of islands practically right off their doorstep yet not in their possession, has been irksome to the nation for a long time.
Added to that is the dispute over the discovery of the islands and subsequent domination of them. For example the Wikipedia pages in English and in Spanish differ slightly in who took possession of them, when and how, stressing different but equally valid viewpoints. In all fairness it needs to be said that the islands changed hands multiple times from French, to British, to Spanish and even US hands -- often in an economic context -- leaving open various interpretations.
Painting of Argentinian Gauchos on the Falkland plains, painted by Royal Navy Admiral Fanshaw
This dispute has always led to tensions, but none as strong as after the United Nations passed a resolution on decolonisation in the 1960s which Argentina interpreted as favourable to its position. A peaceful and symbolic occupation of the islands by Argentine civilians took place in September 1966, known and celebrated in Argentina as Operation Condor (an event rarely mentioned for example in English historical literature about the islands). In the following years Argentinean governments after governments attempted to assert their position, and public opinion grew stronger in favor of an Argentine Malvinas. Many people born in that time period grew up believing the British colonial occupation was unjust.
By the 1980s Argentina was ruled by a political junta after Peron's death, introducing a time of political turmoil and military dictatorship, collectively known as the Dirty War. At the height of its rule, the junta under Leopold Galtieri resorted to everything in their power to maintain control over the nation, including the shutdown of Congress, removal of the Supreme Court, banning political parties and mass murder of any unwanted subjects.
Amids growing unrest within its ranks, the junta decided on one final blow to maintain power: pushing the patriotic button and unite the nation under a shared cause. And there was only one patriotic element that all Argentinians shared, regardless of political affiliation -- that the Malvinas should be Argentine.
Faceoff: Margaret Thatcher and Leopold Galtieri
The Falkland War
On 2 of April, 1982, Argentine military units took over the islands. After less than 3 months occupation, they lost the islands to a British expeditionary force. This loss weakened the junta even more, causing its eventual ousting in 1983 and spelling the end of the Dirty War, while strengthening Thatcherism in Britain.
While another invasion anytime soon is unlikely (especially considering the massive military force Britain left behind on the islands), tensions have remained in place between the two nations to this day.