What Planck did to get the Planck length is a clever bit of calculation. Firstly he considered some fundamental constants of nature.
General relativity is concerned with two constants of nature: G (the gravitational constant, which determines the strength of gravity) and c (the speed of light).
Quantum field theory, on the other hand, is concerned with c (the speed of light) and h (Planck's constant, which determines the amount of uncertainty in our knowledge).
Physicists are attempting to unite these theories into one theory of quantum gravity. If this happens, we can expect all three of these constants (G, c and h) to play a role in the new theory.
Now note that:
The units of G are (distance * distance * distance) / (mass * time * time) The units of c are distance / time
The units of h are distance * distance * mass / time
Planck observed that there is only one way to combine these constants to obtain a distance:
Distance = square root (h*G/c*c*c)
The resulting distance is called the Planck length.
It is a curious observation that Planck made: there is no other way these constants can be put together to get distance. This means that any theory using only these three fundamental constants to predict distance (as a theory of quantum gravity would have to) will give a distance that is this length multiplied by a numerical constant. This gives the Planck length its meaning, because physicists suspect that it is the distance at which the effects of quantum gravity will become apparent and may even be the smallest meaningful length that exists.
Answered by:
Sally Riordan, M.A., Management Consultant, London
'The mathematician's patterns, like the painter's or the poets, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'