This artist's concept shows a celestial body about the size of our moon slamming at great speed into... [+] a body the size of Mercury. NASA's Spitzer Space Telescope found evidence that a high-speed collision of this sort occurred a few thousand years ago around a young star, called HD 172555, still in the early stages of planet formation. The star is about 100 light-years from Earth. Image Credit: NASA/JPL-Caltech
Why don't planets orbiting the Sun ever hit each other?
They did! The current solar system is the stable aftermath of a fairly violent growth period, so what we see now is the result of a several-billion-years-long evolution.
Many of the objects in the universe didn’t begin their existence in the same form as they are seen now- large objects are usually built up from much smaller objects, instead of appearing rapidly at full size. This holds for both galaxies and planets. The planets in our solar system started out just as slight clumps of dust and gas inside a larger disk of more gas and dust. There’s no particular reason for a fixed number of these clumps to pop up, and we fully expect that there were lots of them.
This artist's concept shows a very young star encircled by a disk of gas and dust, the raw materials... [+] from which rocky planets such as Earth are thought to form. Image credit: NASA/JPL-Caltech
With lots of little clumps, each one of them could start running into the surrounding gas & dust, and some of that dust might have stuck to each little clump. This means that we soon have a very crowded solar system of lots of growing pebbles, growing by running into more dust. Each of these little things might become a planet if they keep on collecting more material, but this method of collecting the stuff around them isn’t very efficient. The star is also interfering here, gradually blowing away all of the leftover gas that hasn’t gone into making the star or making our little planet candidates. But there’s a much faster way to grow - collisions.
If each of these things smashes into another one, they can double their size very rapidly; and our current understanding of the early solar system suggests this is precisely how these things grew to roughly planet sized objects. The planet candidates which grew the fastest the earliest could continue to grow to a very large size, since they could capture smaller objects gravitationally, eventually pulling them into a collision course. Incidentally, this gravitational capture is how we think Mars got its two moons - Phobos orbits so close to Mars that it will eventually either break apart or smash into the surface of Mars.
Our moon is also thought to be the remnants of one of these collisions between almost-full-planets. Known as the Giant Impact Hypothesis, the idea suggests that an object about the size of Mars smashed into the proto-Earth, and some of the material slung out from the Earth’s surface recombined into the Moon. This is still ancient history in terms of the lifetime of the Earth- the Earth began forming around 4.6 billion years ago (give or take a few tens of millions of years), the impact would have taken place about 20 million years later, when the proto-Earth was already pretty much formed.
This outstanding view of the full moon was photographed from the Apollo 11 spacecraft during its... [+] trans-Earth journey homeward. When this picture was taken, the spacecraft was already 10,000 nautical miles away. On board Apollo 11 were commander Neil Armstrong, command module pilot Michael Collins and lunar module pilot Buzz Aldrin. While astronauts Armstrong and Aldrin descended in the lunar module Eagle to explore the moon, Collins remained on the command and service module Columbia in lunar orbit. Image Credit: NASA
There was also a period of time called the Late Heavy Bombardment in the solar system’s history, where all the solar system bodies, including the moon, got pelted with a fairly heavy stream of asteroids. This started around 4 billion years ago and continued for about 300 million years. We can still see the marks of this on the Moon - the dark mare on the moon date to the Late Heavy Bombardment.
We’re still getting hit with stuff flying around in our solar system - it just happens that the majority of it is pretty small (on a cosmic scale) to have survived this long without already having crashed into another object. These “small” objects are still sometimes big enough to be of concern to life- the death zone surrounding the impact just depends on how big that object is.
All of the large planets have settled into stable orbits that don’t interfere with each other, after getting through that first 20 million years of chaos, so it’s very unlikely that the large planets in our solar system will crash into each other until the dynamics of our solar system change.
The moon does not fall to Earth because it is in an orbit.
One of the most difficult things to learn about physics is the concept of force. Just because there is a force on something does not mean it will be moving in the direction of the force. Instead, the force influences the motion to be a bit more in the direction of the force than it was before.
For example, if you roll a bowling ball straight down a lane, then run up beside it and kick it towards the gutter, you apply a force towards the gutter, but the ball doesn't go straight into the gutter. Instead it keeps going down the lane, but picks up a little bit of diagonal motion as well.
Imagine you're standing at the edge of a cliff 100m tall. If you drop a rock off, it will fall straight down because it had no velocity to begin with, so the only velocity it picks up is downward from the downward force.
If you throw the rock out horizontally, it will still fall, but it will keep moving out horizontally as it does so, and falls at an angle. (The angle isn't constant - the shape is a curve called a parabola, but that's relatively unimportant here.) The the force is straight down, but that force doesn't stop the rock from moving horizontally.
If you throw the rock harder, it goes further, and falls at a shallower angle. The force on it from gravity is the same, but the original velocity was much bigger and so the deflection is less.
Now imagine throwing the rock so hard it travels one kilometer horizontally before it hits the ground. If you do that, something slightly new happens. The rock still falls, but it has to fall more than just 100m before it hits the ground. The reason is that the Earth is curved, and so as the rock traveled out that kilometer, the Earth was actually curving away underneath of it. In one kilometer, it turns out the Earth curves away by about 10 centimeters - a small difference, but a real one.
As you throw the rock even harder than that, the curving away of the Earth underneath becomes more significant. If you could throw the rock 10 kilometers, the Earth would now curve away by 10 meters, and for a 100 km throw the Earth curves away by an entire kilometer. Now the stone has to fall a very long way down compared to the 100m cliff it was dropped from.
Check out the following drawing. It was made by Isaac Newton, the first person to understand orbits. IMHO it is one of the greatest diagrams ever made.
What it shows is that if you could throw the rock hard enough, the Earth would curve away from underneath the rock so much that the rock actually never gets any closer to the ground. It goes all the way around in the circle and might hit you in the back of the head!
This is an orbit. It's what satellites and the moon are doing. We can't actually do it here close to the surface of the Earth due to wind resistance, but on the surface of the moon, where there's no atmosphere, you could indeed have a very low orbit.
This is the mechanism by which things "stay up" in space.
Gravity gets weaker as you go further out. The Earth's gravity is much weaker at the moon than at a low-earth orbit satellite. Because gravity is so much weaker at the moon, the moon orbits much more slowly than the International Space Station, for example. The moon takes one month to go around. The ISS takes a few hours. An interesting consequence is that if you go out just the right amount in between, about six Earth radii, you reach a point where gravity is weakened enough that an orbit around the Earth takes 24 hours. There, you could have a "geosynchronous orbit", a satellite that orbits so that it stays above the same spot on Earth's equator as Earth spins.
Although gravity gets weaker as you go further out, there is no cut-off distance. In theory, gravity extends forever. However, if you went towards the sun, eventually the sun's gravity would be stronger than the Earth's, and then you wouldn't fall back to Earth any more, even lacking the speed to orbit. That would happen if you went about .1% of the distance to the sun, or about 250,000 km, or 40 Earth radii. (This is actually less than the distance to the moon, but the moon doesn't fall into the Sun because it's orbiting the sun, just like the Earth itself is.)
So the moon "falls" toward Earth due to gravity, but doesn't get any closer to Earth because its motion is an orbit, and the dynamics of the orbit are determined by the strength of gravity at that distance and by Newton's laws of motion.
note: adapted from an answer I wrote to a similar question on quora