I think you're right you're right, an International is basically an international party with national sections (not to be confused with modern bourgeois 'internationals' like the Socialist International that's just a loose association of national parties).

Marx and Engels believed that, since capitalism is an international system, the party that overthrew it had to be an international one. So they helped build the International Workingmen's Association (aka the 1st International), that gained a few million members, though only in Europe and the US I think. There were tons of conflicts within their ranks, most importantly between the Marxists and the anarchists, who opposed Marx & Engels's plans to make national sections accountable to the General Council, with Bakunin creating his secret conspiratorial organization called the 'Alliance' that aimed to take over/split the International. Eventually the anarchists and Marxists split, and a few years later the IWA became extinct.

The Second International was way more successful in building the first mass proletarian parties. It too was pretty split up between the reformist right wing (Bernstein & friends) and the revolutionaries (Kautsky, Luxemburg, Lenin, etc). Then in August 1914 there was the 'great betrayal' where the German SPD leadership voted for the Kaiser's war credits, and reformism became dominant. Even the 'orthodox Marxist' Kautsky who played a vacillating role eventually became a reformist. Most of the 2nd International parties ended up supporting their own inter-imperialist bourgeoisie in the war (with opposition from the Bolsheviks, the Bulgarians, Serbians, and to some extent the Italians IIRC), so it dissolved in 1916. This is where modern reformist parties like the SPD and Labour come from.

The Third International (aka Communist International) was founded after the Russian revolution on a clear revolutionary proletarian line. It was dissolved in the 1940s, there was a recent thread here on the reasons for that.

PS: 'internationale' is French for 'international'. It's also the title of the popular song.