Firstly, distance is used for focal length because it measures the distance between the plane of the lens and the point at which refracted rays meet at a point, when the incident rays were parallel. Below is a simple diagram of a single lens. Note: This is only for convex lenses.

The use of millimetres is simply because it is a scale appropriate for this measurement. i.e. the most extreme lengths don't become numbers that are too large or small for us to comprehend easily. Theoretically any measure of length or distance can be used, but this becomes impractical. For instance a 50mm lens could also be said to be approximately 5.28511705 × 10^-18 light years or 0.0005 Km. Both of those measures are extreme but valid, although not practical.

Why not centimetres? Many lenses have focal lengths that are not whole centimetres, and if possible it is better to represent a number without decimal points, and so mm is a more practical unit. There is almost certainly a historical/traditional reason as well.

Camera lenses work on the same principle as the simple single lens, but include many elements for focusing and telephoto purposes.