If there was a world of air and clouds with the only solid floating then how would this be? What are the possible ways for these islands to float. This is assuming they were almost like small, floating tectonic plates.
Edit: Someone did give an answer that included a dense core with the plated orbiting it. How would this be possible. Maybe there are super dense clouds around this core also pulling this down. What is the science?
Things float because they are (on average) less dense than the medium they float in. Any other means of staying on top is not floating but flying or hovering, which requires energy to counter gravity.
So, the simple, albeit fairly impossible, Answer is that the air your islands float in is extremely dense.
I assume you want some kind of breathable atmosphere on the upper side of your islands.
That would mean a two-phased atmosphere: your incredibly dense floating-gas below, and above that your regular breathable and very much non-dense gas above that.
That means the density cannot be achieved by pressure (which would eventually liquefy the gas anyway), because that would apply to the air above too, but it must be achieved by the molecule mass. Something like gaseous Uranium.
This is possible, but it would be a bit on the warm side.
At least 4131°C, to be precise.
Still cooler than the surface of the sun (approx. 6000°C), but still...
At these temperatures it is no longer relevant if an atmosphere was theoretically breathable, because the people who could report breathability have long turned gaseous themselves.
Or, to put it short: It can't be done.
EDIT
Let's take a closer look at gases.
Under standard conditions (approx: 1 bar pressure, 20°C), WF6 (Tungsten hexafluoride) is the densest gas, with a density of 12.4g/l.
The density of water is 1000g/l. The density of humans is somewhere in the same ballpark (depending on body fat, the volume of air currently in the lungs, ...).
Rock has densities between 1200g/l and 3300g/l, depending on it's composition.
As you can see, your island is some two orders of magnitude more dense than any conceivable gas. They could float on liquids, though, if you tune things just right.
In Weiss and Hickman's "Deaths Gate Cycle", they had a coral like creature that built calcite foam shells around hydrogen bubbles, and then a lot of handwavium to avoid questions like "where did the calcium originate? " and is this mathematically feasible? "
I can do something close, but yet entirely possible within our known sciences.
In 1984, Larry Niven published a book called The Integral Trees. The setting is a vast gas torus, around a neutron star. The gas torus is a known object, and Niven did the math to make the setting scientifically accurate. This gas torus was formed by a gas giant's atmosphere being sucked into orbit around the neutron star.
The gas torus was located 26,000 kilometers away from the neutron star, but, after a detailed examination of the numbers- distances, the neutron star's size, orbit time, etc, it was found that it was incorrect. But by adding a zero to the kilometers, it was corrected. So, Niven just dropped a zero, and removed public interest in this fantastical object.
So we have a place for floating islands, but no floating islands. And that's where Jupiter's Trojans come in. They are two large clusters of asteroids at Jupiter's L4 and L5 points. These asteroids are numerous, and there are estimated to be around 600,000 asteroids in between 2km and 1km in the L4 group. In the L5 group, there are over 1 million asteroids above 1km in size.
Now, we just put these two things together. It is not a world in the strictest sense, but no world seldom is. The main problem is the absence of gravity, but we don't know what happens when someone is born in zero gravity. That means you can make something up for yourself. There is a miniscule amount of gravity on these asteroids, but it's so negligible it could just be called zero gravity. Plants should still be able to grow downwards on their own. They should be extremely tall, but I cannot find anything about that. This should be enough for your world, as anything else would be not be your work.
Look into something like Bioshock Infinites floating city of Columbia. In the world your building the core of the world can be a super powerful magnet, and the bottom of your 'islands' can be made out of some strange superconductor that doesn't rely on cold.
