How is an archipelago (group of islands) formed?

It is formed over time. As a tectonic plate moves over a volcanic 'hot-spot', from time-to-time volcanic eruptions push lava out of the earth onto the sea-bed and eventually make a big enough pile to form an island. The volcanic activity then stop for a while, the tectonic plate continues to move across the globe and so a different part of the plate is then above the 'hotspot'. When the volcanic activity begins again, a new island (may) be made either very close or some further distance from the previously constructed island(s). Thus you see a kind of 'string' effect where Island chains (Archipelagos) are created by this volcanic activity.