Watch the air thin as you climb

Density altitude is pressure altitude corrected for non-standard temperature — the altitude at which the air density you're flying in would occur in the standard atmosphere. On a hot day at a high field, density altitude can be thousands of feet above your actual elevation, which is why aircraft climb and accelerate so poorly in those conditions.

First find pressure altitude: PA = field elevation + (29.92 − altimeter setting) × 1000. Then correct for temperature: DA = PA + 120 × (OAT − ISA temperature), where ISA temperature is 15 °C minus about 2 °C per 1,000 ft of pressure altitude. The calculator above does this for you.

As air pressure drops, the partial pressure of oxygen drops with it, so each breath delivers less oxygen to your blood. Hypoxia can begin degrading night vision as low as ~5,000 ft, and judgment well before you'd notice. Under FAR 91.211, U.S. pilots need oxygen above 12,500 ft cabin altitude (after 30 minutes), continuously above 14,000 ft, and must provide it to passengers above 15,000 ft.

For an unacclimatized person breathing cabin air, SpO₂ falls from ~97% at sea level to roughly 87% at 10,000 ft, ~72% at 18,000 ft, and ~50% at 25,000 ft. The drop accelerates because the oxygen–hemoglobin curve falls off a cliff once arterial oxygen pressure drops below ~60 mmHg.

A normally-aspirated piston engine loses power roughly in proportion to air density — about 3% per 1,000 ft of density altitude (Gagg–Ferrar). At 8,000 ft density altitude a 180 hp engine makes closer to 135 hp. Turbocharged engines hold rated power up to a critical altitude, then fall off the same way.

Yes — it's a myth that they don't. Jet engines produce thrust (not horsepower), and thrust falls as you climb because thinner air means less mass flowing through the engine — roughly in proportion to the ambient pressure ratio, so a turbofan at FL400 may make only a third of its sea-level static thrust. Turboprops produce shaft horsepower and are usually flat-rated to a critical altitude (like a turbocharged piston), then lose power as density falls. Jets cruise high not because they keep their power, but because drag drops even faster up there and fuel efficiency improves — so the reduced thrust carries them further per pound of fuel.

The fuselage can only hold a limited pressure difference (ΔP) between inside and outside. At cruise the system maintains that maximum differential, which works out to a cabin altitude near 8,000 ft for most airliners. Newer composite jets like the 787 and A350 tolerate a higher ΔP and target ~6,000 ft, which reduces fatigue on long flights.