I'm not known for simple explanations, but I'll take a shot. We'll use the example of you taking a picture of Mickey Mouse.
The focal length of a lens determines how big Mickey looks in the picture. The bigger the focal length number, the bigger your Mickey looks. If you have a zoom lens (one that has a range of focal lengths, like an 18-55mm lens), you can zoom in on Mickey. When you do that, you are increasing the focal length of the lens. Zooming in makes Mickey look bigger, but it may also mean that you can't fit all of him in the picture. Zooming out makes him look smaller and shows more of him. Using the example of an 18-55mm zoom lens, 18mm is shortest focal length (most zoomed out) and 55mm is the longest focal length (most zoomed in).
The focal length isn't the only thing that tells you how zoomed in you are. The other thing you need to know is how big your sensor or your film is. The smaller that is, the more zoomed in everything looks. If your camera has a sensor the size of a tic-tac, a 50mm lens magnifies things a lot. On the other hand, if you are shooting on a piece of film the size of a sheet of paper, a 50mm lens makes things look very small.
To understand why, let's go back to our Mickey example and compare some real-world cameras. Let's look at the Canon S5, the Canon Rebel with a 50mm lens, and the Canon 1Ds. Let's assume that they are all using a 50mm lens or a zoom lens set to 50mm. If you could get inside the camera and put your eye where the sensor is, you would see the exact same picture on all three cameras. The focal length is what determines how zoomed in you are and a 50mm focal length is the same regardless of the camera.
The difference, and there is a big difference, is that the sensor is a different size in each of the three cameras. On the 1Ds, the sensor is very large and can see much more of the picture coming in from the lens. On the Rebel, the sensor is smaller and can only see about 60% of the picture coming in from the lens. On the S5, the sensor is smaller still and can only see about 15% of the picture coming in from the lens.
Now let's assume that you want to make a 4x6 print of the Mickey picture from each camera. With the 1Ds, all of Mickey fit onto the sensor and so you have a full body picture of Mickey. With the Rebel, just the upper half of Mickey fit onto the sensor, so you have the same picture as with the 1Ds, only it looks zoomed in more. With the S5, only Mickey's face fiton the sensor, so the picture looks like you zoomed in on his face. In all three cases, you used the same focal length, so the picture the lens made was the same. The difference is that the smaller sensor only saw a smaller portion of that picture and so it looked more "zoomed in."
If you actually saw the picture as it was hitting all three sensors, you would say that the smaller sensors were actually chopping off or missing parts of the picture rather than zooming in. However, we don't look at pictures on sensors. We look at prints (or screen images) and we make those about the same size regardless of the sensor size. Because of that, cameras with smaller sensors make pictures taken with the same focal length look more zoomed in.
Back in the days of 35mm film, almost every hobbyist photographer used a 35mm film camera. The film was the sensor and it was the same size for everyone. People got used to what a picture from a 50mm lens looks like on a 35mm film camera. There really isn't anything magical about the size of 35mm film; it's just what we were used to. Now that people use cameras with sensors of all different sizes, it's harder to compare focal lengths. That 50mm lens from a 35mm film camera makes pictures that look the same on a "full frame" camera (full frame just means the sensor is the same size as a piece of 35mm film), but it looks more zoomed in on a Rebel. If you could use that lens in place of the lens on an S5, it would look even more zoomed in. Because of that, people often talk about focal length equivalent.
The focal length equivalent is a way to adjust for the sensor size. When Canon says that the S5 has a 36mm-432mm focal length equivalent lens, what they mean is that it has the same mangification range that you would get with a 35mm film camera using a 36mm-432mm lens. It really has a 6mm-72mm zoom lens but the sensor is 1/6 the size of a piece of 35mm film.
You also hear things like 12x zoom. That doesn't mean that things look 12 times bigger. That means that the longest focal length is twelve times bigger than the shortest focal length. For the S5, that's from 72mm down to 6mm (432mm to 36mm equivalent).
You can't really tell how close a camera can zoom by looking at this number because it doesn't tell you where it starts. For example, I have two 3x zoom lenses for my DSLR. One goes from 24mm to 70mm and the other goes from 70mm to 200mm. They are both roughly 3x (70/24 and 200/70), but they cover different focal length ranges.
Well, I've wandered all over and lost the two year old a long time back. How about: Focal length is how zoomed in a lens is. Bigger numbers mean more zoomed in. Smaller cameras zoom in more than bigger cameras using the same focal length.