Ghostman describes mostly volcanic islands. Which are more of an exception. For a lot of the worlds islands, it helps if you stop thinking about them as if they where islands, and start thinking like they where mountain-peaks, which is exactly what they are.

The relief of our planet doesn't 'end' at the sea. The sea is just an alternate filling, opposed to air. Sometimes there are more or less 'lone' mountain peaks on land, which means those also exist underseas. Sometimes they are so high, not all of it is covered under water. That is an island.

There are in fact other ways for islands to come into being. If I recall correctly, the Canary Islands are in fact located on top of a kilometres thick stack of shells of dead sea-animals.

Along the coast, islands often come into being from sedentary processes. Rivers corrode mountains, this corrosion floats downstream, and at the rivermouth into the sea, this sediment gets dumped, if enough is dumped, islands form. Sediment-islands usually get shaped by water currents.

I hope this is useful