How do airplanes and ships stay in the air/float?

well i cant answer about the airplanes but as far as the boats they have to displace their weight in the water that keeps them afloat

Archimedes' principle

for airplanes it is a combination of airspeed and lift. Lift is achieved by the shape of the wing. The cross section of an airplane wing is tear drop shaped (sort of). When there is proper airspeed, the shape of the wing is such that the airflow over and under the wing create different air pressure. The air pressure under the wing is greater than the top causing the airplane to rise.

For airplanes-- This will help you grasp an understanding: take a sheet of paper and blow over the top. Notice how it rises? Imagine this times 1000 with the accompianment of engines actually pulling air towards the wings. For ships-- Anything with a density less than the substance it is put into will float. Metal by itself, of course, will sink in water, but metal along with a great amount of air will be enough to float in water.

Boats float b/c they displace a greater weight of water then they weigh. The water they take up is greater than the amount they weigh. Airplanes fly b/c the air flowing over the wings is disturbed. As a result the pressure atop the wing is less than that below the wing and lift is created

The real reason an airplane flies has to do with pressure. As the wing travels, it "cuts" the air, forcing some of it above and some of it below. All the air travels over the wing in the same amount of time, but the air on the top of the wing has a longer way to go (bc of the wing's shape), and so it travels more quickly. Pressure is an empyrical measure of the kinetic force imparted to a two-dimensional surface in a given amount of time by the impact of countless particles (usually gas particles). The air particles moving over the top of the wing have less time in which to strike the wing -per unit of area-, and consequently will impart less kinetic energy into the wing than their partners on the reverse side. This results in a pressure differential, but really it's a difference in forces being exerted on the top and the bottom of the wing, which causes the wing to be forced upward, viz. lift.