Definition
Buoyancy is the upward push on an object that sits in a fluid, which is any liquid or gas that can flow, like water or air. It's why a heavy object feels much lighter when you lift it underwater than in the air: the fluid around it is pushing it up.
What happens when you push a pool float under the water
Have you ever pushed a rubber ball or an empty plastic bottle deep under the water in a bathtub? The harder you press, the more you feel something pushing back up against your palm. The moment you let go, the ball shoots up and pops out of the water.
That upward push on the object is buoyancy. An object underwater gets squeezed by water pressure from every side. But water pressure gets stronger the deeper you go, so the pressure pushing up from below is always bigger than the pressure pressing down from above.
Because of that difference between the bottom and the top, the object ends up with a net force lifting it upward. It's also why you can scoop up a friend in a swimming pool so easily. The water is helping you lift from underneath.
This doesn't only happen in water. Air is a fluid too, so a tiny, invisible buoyant force is always acting on a floating balloon, and even on your own body.
A bit more precisely: Archimedes' principle
Fill a bathtub to the brim and climb in, and water sloshes over the edge. To make room for itself, an object has to push aside as much water as its own volume.
The ancient Greek mathematician Archimedes wrote down the key rule for this in his book "On Floating Bodies". The size of the buoyant force is exactly equal to the weight of the fluid the object pushed aside. Physics calls this Archimedes' principle.
For example, if you fully submerge an object with a volume of 1 liter, it pushes aside 1 liter of water, which weighs about 1 kilogram. The water then pushes that object up with a force equal to the weight of 1 kilogram (about 9.8 newtons).
It's as if the water that got shoved out of the way is shoving back, saying "give me my spot back!" The heavier the displaced water, the bigger the buoyant force, in direct proportion.
Why doesn't a giant steel ship sink?
Toss a small steel ball or a coin into water and it drops straight to the bottom. So how can a huge steel cruise ship or container ship, far heavier than any steel ball, float safely on top of the water?
The secret is the overall shape and the density of the object. Density is how tightly the material is packed together. A ship isn't solid steel inside. Most of it is designed to be empty, filled with air. The key is that the hull is built as a wide, bowl-like shape, a structure that can push aside an enormous amount of water.
If the most seawater the ship can push aside weighs more than the whole ship, steel and air together, the ship floats. It sinks only until it has pushed aside water equal to its own weight. On the other hand, if the water an object pushes aside weighs less than the object itself, the object sinks to the bottom.
Submarines use the same principle. When the tanks inside fill with seawater, the sub gets heavier and sinks. When compressed air pushes the water back out, the sub gets lighter (the buoyant force becomes greater than its weight) and glides back up to the surface.
Common misconceptions
Heavy things always sink, and light things always float.
Whether something floats or sinks depends not on weight alone but on its weight for its size (density) and how much fluid it pushes aside. Even a steel ship weighing tens of thousands of tons floats easily once its inside is hollow and its volume is large, because it pushes aside so much water.
Where you meet it
When you lie back in a pool and relax, your body floats. That's buoyancy at work.
Helium balloons at an amusement park and hot-air balloons in the sky also rise thanks to the buoyant force of the air pushing them up.
In one sentence
Buoyancy is the upward push on an object in a fluid, equal to the weight of the fluid the object pushes aside.