Submerge anything and the fluid above the top face presses down while the fluid below the bottom face presses up — and because pressure grows with depth, the upward press always wins. The difference is the buoyant force: $F_b = \rho V g$, read aloud F-b equals rho V g. $F_b$ is the upward force in newtons, $V$ is the volume of fluid displaced — for a fully submerged body, simply its own volume — in cubic metres, $g$ is 9.81 m/s², and $\rho$ is the density of the FLUID, in kg/m³.

That last point carries the whole lesson, so say it out loud: the density in Archimedes' relation is the fluid's, never the body's. A steel block and a wooden block of identical volume, both held fully under the same water, feel exactly the same buoyant force. They behave completely differently when you let go, but that is a contest between the buoyancy and their own weights — and the buoyancy itself never knew what it was lifting. Reaching for the body's mass here is the most common mistake in this subject, and it is worth catching before it becomes a habit.

Float or sink then reduces to a single comparison. Put both densities on water's scale as specific gravities, $SG = \dfrac{\rho}{\rho_{water}}$, and the rule is one line: if the body's SG is less than the fluid's it floats, settling until the volume it has pushed aside weighs exactly what the body weighs. If it is greater, then even fully submerged it cannot displace its own weight, and there is no more volume left to buy buoyancy with. Down it goes.

Two nuggets on the way out. Seawater at SG 1.025 is denser than fresh, so a ship rides higher at sea than in a river — which is exactly what the Plimsoll marks on a hull are for. And a steel ship floats not because steel is light but because the hull encloses a great deal of air, and it is the SHIP's average density, not the plate's, that the water is asked about.