Satellite
A satellite is any object that orbits a larger body: natural satellites like the Moon, or artificial ones launched for communication, navigation, and science.
Definition
A satellite is anything that orbits a planet or other larger body under gravity: the Moon is Earth's natural satellite, and people have also launched thousands of artificial satellites to take pictures, relay signals, and help devices find their location. A satellite stays up through a balance between its forward speed and gravity's pull, and its altitude sets both its speed and its period, with higher orbits meaning slower motion and a longer time to go around. For a circular orbit of radius $r$ about a mass $M$, speed works out to $v = \sqrt{GM/r}$ and period to $T = 2\pi\sqrt{r^3/GM}$, and orbits are classified by altitude and inclination: low Earth orbit below about $2{,}000\ \text{km}$, medium Earth orbit, geostationary orbit at $35{,}786\ \text{km}$ where $T$ matches a sidereal day, and sun-synchronous polar orbits that precess to keep constant lighting. Low orbits decay over time from atmospheric drag, and satellite clocks must be corrected for both special and general relativity to stay accurate.
Example
The International Space Station is a satellite where astronauts live, orbiting about $400\ \text{km}$ up at $7.7\ \text{km/s}$ with a $93$-minute period; GPS satellites orbit at about $20{,}200\ \text{km}$ with a $12$-hour period, and communication satellites at $35{,}786\ \text{km}$ take exactly one day, so they hang over the same spot on the equator. GPS satellite clocks run about $7$ microseconds a day slow from their orbital speed under special relativity, but about $45$ microseconds a day fast from sitting in weaker gravity under general relativity, for a net gain of about $38$ microseconds a day. Left uncorrected, that drift would throw off computed positions by roughly $10\ \text{km}$ every day.
Key Insight
A satellite stays up with no engine running at all: it moves sideways so fast that as it falls toward Earth, the ground curves away beneath it just as quickly, so it never lands. The first artificial satellite, Sputnik 1, was about the size of a beach ball when it launched in 1957, and thousands of working satellites now follow the same orbital laws Kepler once worked out for the planets. In a real sense nearly every satellite is a working test of gravity, and the GPS constellation is the most widespread everyday application of Einstein's relativity: turn-by-turn directions on a phone work only because the physics of orbits and curved spacetime is built directly into the system.