Formation of Iceland
The answers so far are broadly correct. Iceland has formed from a coincidence of a constructive plate margin and a mantle plume, or "hot spot". But to really understand why Iceland is there we need to look at a normal constructive plate boundary.

A normal mid-ocean ridge is formed where two plates spread apart so that the mantle, which is at a certain temperature, rises upwards and experiences a decrease in pressure. The lower pressure together with the temperature of the mantle allows some (not all) of the mantle to melt - a process called partial melting. Only a small proportion of the mantle actually melts. This melt rises up through the overlying crust in a series of magma bodies, dykes, and lavas, to form the crust. Crucially, the thickness of the crust that is formed depends on the amount of melt that is created, referred to as the melt thickness.

A normal mid-ocean ridge creates a very thin crust - usually about 7 km. That's just 7 km between the ocean and the mantle. But Iceland is not normal. It sits above a mantle plume, which is a region of anomalously warm mantle. So what happens when we depressurise an area of mantle at a higher temperature than normal? We create a much greater melt thickness. This means the crust that is produced above the plume is greater - as much as 30 km. Since the icelandic crust is so much thicker than normal oceanic crust, the upper part of it is able to rise above the level of the sea. And thus Iceland exists.

To answer your followup questions:


  Why do not mid ocean ridges often rise above the sea surface to form islands?


Because it's very rare for a hot spot to coincide with a mid-ocean ridge.


  Why are islands more common along convergent plate boundaries where they often form archipelagos?


Because these islands form through very different processes. Convergent plate margins introduce water and other volatiles into the mantle, and these have the effect of reducing the melting temperature of the mantle. This produces melt, which rises up and forms volcanoes and islands. This process occurs at virtually every single destructive plate boundary because the subducting oceanic crust contains water which is liberated during subduction.