If an entire planet was made up of nothing but small archipelagos, what would come of it? Would living things be able to exist? Would said living things ever form cities or even nations? How would I explain why the world is nothing but small islands in a massive ocean?
Continents are formed by tectonic activity, so you would need a planet with little or no tectonic activity. No large moons either. The planet would need to be very old and/or having formed from a nebula poor in radioactives, which means a Population II or (more rare) Population III star. Elements beyond iron would be rarer than on Earth.
Life would be different from Earth's (I believe that several heavier-than-iron elements are required by Earth metabolisms, even if in trace amounts), but CHONP would be there, and the basic organic molecules could form. So, I don't see why there shouldn't be life on such a planet.
Of course it would reman aquatic for way longer than on Earth.
But at some point, islands could be formed through biogenesis - you'd have a world of shallow oceans and coral reefs.
There is always the possibility that the planet once had continents. At one point Earth had 1 big continent and shallow seas. The archipelago planet could be a several million year phase the planet is going through.
Another possibility is that it is a water world with or without plate tectonics. It is theorized that before Earth got hit by the planetoid that created the Moon, Earth was a water world. Volcanoes would likely happen even without plate tectonics, just no ware near the rate we have on Earth. And coral reefs are also a possibility.
I would bet that in the second scenario, any intelligent live would be amphibians. Too much tech involves fire. Having islands would allow for smelting and most of the tech that we know.
In the first scenario, I don't know that there would be time for intelligent life to form unless it represents the survivors of whatever caused the continents to disappear.
There are different theories regarding plate tectonics and why it exists. These theories try to explain why Earth has plate tectonics and Venus, and likely Mars don't have it. Consensus seems to be that an Earth-like planet should have planet tectonics, and consequently, continents, but we shouldn't take it as settled science. For Earth itself plate tectonics took more than a billion years to establish itself, and before that Earth is thought to be pretty much what you have described - an Archipelago Planet.
Aquatic life should have no problem developing on such a planet. However, terrestrial life would have major problems. Not only the amount of dry land would be small - this land would be split among large number of isolated pockets, and any species that are not good swimmers or flyers would have to die out when an island goes under. I propose that your planet would have a civilization of mermaids :)