Archimedes, properly
What the principle actually says, and the part everyone misremembers.
Archimedes' principle says that any object sitting in a fluid feels an upward push equal to the weight of the fluid it has pushed out of the way, and the part almost everyone misremembers is that this push exists whether the object floats, hangs suspended or sinks straight to the bottom. The principle describes the push itself, and whether the object floats is a separate contest between that push and the object's own weight.
An underwater vehicle has to solve a problem no aircraft faces: it must weigh exactly nothing once it is in the water it is meant to work in. Getting that condition right by feel, without understanding what the principle claims, is how a first hull ends up wallowing on the surface or settling on the bottom of the test tank, unable to hold the middle depth it was built for.
Climbing into a full bath
Anyone who has climbed into a full bath and watched the water rise, or overflow if the bath was filled too generously, has seen the principle at work in the most literal way, because the water pushed up and over the edge is exactly the volume the body has shouldered aside. Weigh that spilled water and it equals the upward push the body feels, which is the whole of Archimedes' claim delivered as a kitchen demonstration.
The bath hides one thing, because a body in a bath is obviously floating. A stone dropped into the same bath displaces its own volume of water on the way down and feels an upward push the entire time. That push is simply too small beside the stone's weight to keep it from reaching the bottom.
Why a steel ship floats
The version taught early, that heavy things sink and light things float, sounds like a statement about weight but is really a loosely worded statement about density. A solid block of steel sinks instantly, while a ship built from the same steel floats, because the ship is a shape enclosing a great deal of empty space, and the whole shape's average density against water's decides the outcome. Archimedes' principle itself stays out of that contest and supplies only the size of the push.
Steel is roughly eight times denser than water, so a steel hull floats only when the water its enclosed shape displaces weighs more than the steel used to build it, which means the hull must displace more than eight times the volume of steel in its plates. Asking how much space a hull encloses relative to its weight, instead of how heavy it is, leads to different decisions. A vehicle shaped for strength first tends to have buoyancy corrected afterwards with bolted-on foam or trimmed ballast, while one whose shape was chosen against a target density from the first sketch rarely needs correcting.
When the water and the hull change with depth
The principle is usually applied with the fluid's density taken as fixed, which suits a bath or a pool but bends once a vehicle goes deep enough that the water around it is colder, saltier or slightly compressed by the weight above. The object's volume is assumed fixed too, which holds well for a rigid steel hull but fails for anything that can be squeezed, such as a pocket of trapped air or a block of syntactic foam, because a shrinking volume displaces less water and loses buoyancy as it descends. The principle still holds at every depth if the true, momentary volume and density are used. The figures measured at the surface simply may no longer apply, and that is the start of a longer story about how a vehicle is held at a chosen depth instead of floating or sinking by accident.