Focal length is the distance from the conversion point to the image plane. The image plane could be film, or a digital sensor.

For the purpose of understanding focal length, think of the lens as being a pinhole that is the focal length distance from the sensor. The further you move the pinhole away (make the focal length longer), the larger the image will be on the sensor.

However, the image also gets dimmer, since the same light that passes thru the pinhole is spread over a larger area. If you double the focal length, then objects in the image double in length and width. That means they occupy 4 times the area in the image. Because the same light is spread over 4 times the area, the result is 4 times as dim. The brightness goes inversely with the square of the focal length.

Now imagine you compensate for the dimness due to double the focal length by making the pinhole larger. (That also makes the image blurrier, but that's not the point here.) You'd have to make the pinhole area 4 times larger, which means you make it's diameter twice the size.

If you've been following all that, you can see that to get a measure of how bright the projection of a lens will be, you take the ratio of the diameter to the focal length. This is exactly what the f-stop numbers are. "f/8" literally means the diameter is the focal length divided by 8. A 50 mm and a 200 mm lens set to f/8 will both project whatever part of the scene they show at the same brightness. The 50 mm will show a wider view projected onto the same sensor. The effective opening in the 200 mm lens will be 4 times in diameter of that in the 50 mm lens.

Real lenses can still largely be thought of as pinholes for the purpose of understanding focal length and aperture. The special thing lenses with all those glass elements do is make a sharp image of the scene at a particular distance, according to how the focus adjustment is set. These lenses give you sharp focus at one distance and worse at others. Pinholes are equally sharp or unsharp over the whole distance range.