Gravity provides the centripetal force to sustain orbits, F= mGM/r2
Gravity keeps satellites in orbit. The closer you are to the Earth, the faster you have to go to maintain your orbit. At low Earth orbit, the altitude of the Space Station, you make an orbit every 90 minutes. At the Moon's distance you need over 27 days to go around the Earth. In-between there is an altitude which matches the rate of the Earth's rotation. Many satellites orbit at this altitude.
GPS satellites orbit the Earth at an altitude of about 20,000 km in medium Earth orbit (MEO). They move in precise paths to ensure global coverage and are spaced evenly to provide accurate positioning data. The satellites follow elliptical orbits that repeat every 12 hours.
Velocity and gravity (movement around the Earth and the gravity of the Earth pulling on it).
The gravity from the sun "holds" the planets into place and make them orbit the sun. Just like the gravity from the Earth makes the moon orbit the Earth.
The main force that causes the moon to orbit around the Earth is gravity. The gravitational pull of the Earth keeps the moon in its orbit, while the moon's own momentum prevents it from falling into the Earth. Additionally, the gravitational pull of the moon also affects Earth's tides.
Satellites need to be out of the earth's atmosphere or the gravity would make them fall to the ground. In space, the satellites are also able to orbit the earth effortlessly because they have less weight.
An object in orbit is falling toward the Earth. Plus, it has some motion 'sideways'. Remember that the Earth is shaped like a ball. The sideways motion of the object is just enough so that the curve of the Earth 'falls away' just as fast as the object falls. So the object keeps falling, but never gets any closer to the Earth.
If they were too far out, then they would not be very useful as weather satellites, because they would orbit too slowly, and they would be too far away to make out the weather.
Yes, a satellite in orbit requires fuel to make adjustments to its trajectory or maintain its position. However, satellites in geostationary orbit can maintain their position without fuel because they orbit above the same location on Earth.
Man-made satellites are either in low, medium or high earth orbits, depending on their function. The closest ones, including the International space station, are in a low earth orbit, up to a height of 2000 km. The International Space Station orbits at a height of around 340km, while the Hubble Telescope is around 595km. The advantage of these is that less energy is required to get there, but the disadvantage is that their orbits may decay over time, since there is some drag on the atmospheric gas still at these heights - so these may need a 'boost' every now and then to maintain their orbits. At higher altitudes, satellites will orbit more slowly, taking longer to make one orbit. The medium Earth orbit extends to 35,786km above the Earths surface, from here the high earth orbit begins. At this point, the time taken for a satellite to orbit once around the earth is 24 hours, the same as one day. 35,786km is a special orbit for Geo-stationary satellites. Since they orbit the earth in the same time that it takes the earth to spin once on its axis, it means that the satellite can stay in roughly the same spot over the earth.
Interia and gravity combine to make a planet stay in an orbit.
Gravity causes a 'mutual' force between every two masses, that attracts each of them toward the other one. The mutual force between the sun and earth is what keeps the earth in its orbit around the sun.