NATO as a body wasn't involved at all (the North Atlantic Treaty is for mutual defence north of the Tropic of Cancer (plus Algeria), whereas the Falklands are a long way further south)

The UN expressed it's concern, demanded an end to hostilities and a complete withdrawal of Argentine forces in the binding security council resolution 502

In terms of individual countries, the EEC and the Commonwealth of Nations (i.e. what used to be the British Empire) imposed economic sanctions on Argentina.

The majority of South America (notably except Chile) supported Argentina (at least vocally).

Chile (who had been through a confrontation with Argentina shortly before) supported the UK, by providing intelligence and warnings of aircraft movements (as well as tying up some of Argentina's best troops, as they were concerned that Chile might intervene militarily over their previous border dispute).

America tried to maintain an appearance of neutrality while supporting the UK, especially turning a blind eye to the Royal Navy helping themselves to fuel and equipment from the joint US-UK base on Ascension Island (it would be hard to object to the British helping themselves to war supplies from one overseas territory while on their way to liberate another). The neutrality was beneficial as it let the US try to act as an arbitrator, until it became clear that the Argentine Junta was telling the US negotiator one thing, only to find that by the time he'd flown to the UK to present it they'd changed their minds, told the American newspapers, and he'd find out their new position from the British when he arrived. Eventually the US came down in support of the UK and provided an assortment of kit to the UK.

The French delayed delivery of Exocet missiles to everywhere in South America, and the British and French intelligence agencies cooperated to buy up any that were on the international market. However, a French technical team that was in Argentina remained finding and fixing faults in Exocets throughout the war.

In what is best described as a botched secret mission Israel sent four airliners of supplies, including Israeli-spec (as opposed to the slightly lesser export-spec) drop tanks for their aircraft, via Peru. The secrecy was sufficiently badly blown that the British embassy was able to get someone to the airport before the planes arrived who was able to take pictures of the Israeli planes being unloaded and the contents being loaded into Argentine planes on the tarmac.

The Argentines also received several planeloads of (mainly Soviet) weaponry from Libya, refueling in Brazil.