The Solar System is bigger than you think.
Before we talk about the Oort cloud, let's take a fast tour through the Solar System. The distances won't be in millions and billions of miles (or kilometers). They will be in astronomical units (AU) - 1 AU is the distance from the Earth to the Sun.
The inner Solar System is from the Sun almost to Jupiter. Then we have the outer planets from Jupiter to Neptune. Jupiter is at 5 AU, five times the distance from the Earth to the Sun, and Neptune is at 30 AU. On Neptune the Sun would look like a bright star.
After Neptune comes the Kuiper belt, including Pluto. Going from 30-50 AU, it's big. At about 50 AU the scattered disk starts and continues to 100 AU. (You can find out about the Kuiper belt and scattered disk in the article "Kuiper Belt - Facts for Kids.")
Then there doesn't seem to be much of interest until the start of the Oort Cloud. No one really knows where it begins, but most astronomers say 10,000 AU. Wow! Not only is that a long way away, but it then stretches to 100,000 AU (or maybe even twice that). This is trillions of miles, about a third of the way to the next star.
Comets have problems surviving, so there must be a reservoir of them in the far distant outer Solar System.
We know that comets run into planets, such as comet Shoemaker-Levy that crashed into Jupiter in July 1994. We also know that they often fall into the Sun. And we know that every time a comet goes near the Sun it loses mass. Comets have many problems surviving and yet there are still comets. Astronomer Ernst Opik - and later Jan Oort - realized this meant that there must be a reservoir (a sort of storage place) of comets far out in the Solar System.
In 1932 Opik suggested that a region of comets surrounded the Solar System a long way out. Most astronomers didn't see Opik's paper, but they did see Jan Oort's paper in 1950. This sphere of comets is usually called the Oort cloud after Jan Oort. The names of things aren't always fair, though some people do call it the Opik-Oort Cloud.
You can see a drawing of what the Oort cloud might be like. In the drawing they slice into the cloud like a cake so that you can see the rest of the Solar System is inside it.
The comets formed with the other Solar System bodies.
Astronomers think the planets and other bodies all formed soon after the Sun did from leftover material. The comets must have formed closer to the Sun than they are now because that's where the material was. However the effects of the gravity of the giant planets either pushed the comets into the Sun or into the outer Solar System.
The Sun's gravity has a very weak hold on the Oort cloud.
Gravity acts over very, very long distances, even though it gets very weak when the bodies are a long way apart. The Oort cloud is so far from the Sun that the comets are also affected by the gravity of other objects in the Milky Way, including other stars. These other objects can occasionally disturb the comets and even knock them into new orbits that bring them into the inner Solar System.
Comets come from both the scattered disk and the Oort Cloud.
The orbits of short-period comets tend to go around the Sun and out to the outer planets, returning in less than 200 years. Their orbits show that they come from the Kuiper belt or the scattered disk. Long-period comets are not so predictable and come from all sorts of different directions. These are the ones that come from the Oort Cloud.
Some of our comets are missing.
There are probably more than a trillion comets in the Oort cloud. However their total mass would probably be about five times that of the Earth. Astronomers think that there should be more of them. It's likely that although a number of comets are pushed into orbits that bring them near the Sun, many more have been knocked right out of the Solar System.
Here is a collection of images about the Outer Solar System.