Argentina invaded and occupied the Falkland Islands. The Falkland Islanders themselves do not want to be ruled by Argentina and appealed to the UK government for help. But a big part of the reason for defending the islands must be political: that Margaret Thatcher had been advised by the armed forces that a war to regain the islands would be easily won by the British side. A successful campain would show her government as a saviour for British subjects which would be a huge politcal advantage for her government. While the war hardly went as smoothly as anticipated, the victory was still generally a poltical success for the Thatcher government. The UK had the option not to defend the islands: e.g. they chose not to oppose the occupation of the Channel Islands by Germany during the Second World War. (Edited this answer to make it clearer that it was actually pretty much a shorter version of Drinking Dano's answer)

Leaving out the long history of who owned the Islands when and at what time, it is a matter of historical fact that Argentina claimed it for its own when it gained independence from Spain in 1816. Then in 1833, the British took the Islands by force and sent the Argentineans packing home. This, as you can imagine, caused a lot of issues between the two countries in the 20th century. The British would have been quite happy to rid themselves of this headache, had it not been the many people of British decent who lived there and had no desire to be part of Argentina. The Juna that had taken over Argentina 1982 and proceeded to run the country in the ground. Many people believe (and I agree) that the Juna decided to start the war as a way to get the people to forget about their trouble and become patriotic about the war. Foolish dictators! The British never really believed that Argentina would invade the Islands and the Governor of the Islands was more then just a little surprised at the build up. In fact, the British only had about 70 Royal Marines on the Islands and one Destroyer. (HMS Endurance) Once Argentina had taken the Island, the British were left with little option besides invading and retaking it. After all, there were many British Citizens there! HMS Invincible and HMS Hermes were sent at once. Shortly after, the British had F-4 Phantoms and Vulcan bombers near by. To make an accedingly long story short, the British retook the Islands with very little trouble from a Military standpoint. There was the loss of HMS Sheffield and some other hardware, but all and all, the Argentineans took much more punishment. While there certainly were political reasons behind the war, (Thatcher did receive a lot of support for it) the ultimate reason for the war itself was (in essence) the United Kingdom had been invaded, via the Falkland Islands.

As pointed out by Drinking Dano, the decision to re- take the Falklands was largely political, Mrs Thatcher winning the general election by a landslide soon after. There is also oil and gas underneath the Falkland islands, although at the moment it is not economically viable to extract it, so the reasons aren't that different to the wars we fight nowadays. I know there is oil and gas under the Falklands as my Father in law owns shares in a Company called 'Falkland oil and gas'. See the link below: http://news.bbc.co.uk/1/hi/world/americas/6501693.stm Most major political decisions are about the money, this one was one of them. North Sea oil is going to run out in about 50 years, we need to find a new supply.

HMS Endurance is not a destroyer, she had no armaments (Apart from small arms carried onboard and a helicopter). She was an Ice breaker/survey ship and was painted Bright red - not the usual warship grey for ships of the Royal Navy

one word my friend, oil

"The Falklands thing was a fight between two bald men over a comb." Jorge Luis Borges Quoted in: http://en.wikipedia.org/wiki/Cultural_impact_of_the_Falklands_War

I was on HMS Yarmouth ( nickname: crazy y ) during the whole conflict and what a lot of people forget is that 252 of my friends lost their lives in a very SHORT period - not like the long outdrawn affair by our men and women in Iraq/Afghanistan. The Argentinians were very brave - especially their pilots. Why - Oh Why do good men and women have to die on all sides of a war - when the Politicians should sort their own problems out amongst themselves.

Britain did not go to war, war was declared on them when a peace of their land was invaded and thier peolpe subjected to a military dictatorship that had previously told the locals to pack thier bags and go. Fighting to take them back was a leagal oblogation as well as the orders from thier peolpe to defend them. Oil became usful later and helps to pay for the military to protect the islands and keep watch over south america.

My there are some cynics out there! The Faulklands were (ARE) important to the British Isles because Britons live there. We went to war to liberate our own people. Although the outcome was good for Thatcher, it was not her motive. Nor was oil or gas. I send a huge thank you to all serviceman involved in that conflict, and this Sunday when I am amongst many others that turn out for a rememberance parade across the country and Commonwealth will remember them that fought, those that died and those injured, their families and again raise an enormous thank you to them all. Our forces may be misused in plenty of areas, (that's a political thing we need to address) but they are OUR forces, OUR men and women ready to defend OUR freedom at all times. We're damned lucky we have them.

It's a matter of principle. No more, no less.

It wasn't the English who went to war with Argentina, it was the British.