The Planck length is sometimes said to be the smallest distance which can be meaningfully measured, or the smallest possible margin of error in a measurement of position. Even those at the forefront of the attempt to understand quantum gravity
do not usually claim a full understanding of its significance, or certainty about the value, but most would guess that it represents the sort of scale at which quantum-gravitic effects should become important in determining behaviour.
To understand this, it helps to see how it is derived.
To start with, our best ways of making positional measurements involve using photons with a very small wavelength. Ones with large wavelengths (like radio waves) give large errors; visible light, with smaller wavelengths, is quite good; but to get very high accuracies we need to use photons with much tinier wavelengths. The smaller the wavelength of the photon, the higher its frequency. The higher the frequency, the higher the energy. (Frequency over wavelength, of course, is a constant - in the case of photons this is c, the speed of light.)
There is a famous equation, E = mc2 implying the equivalence of mass and energy. It turns out that if you want to measure the position of a particle beyond a certain accuracy, you have to use a photon of high enough energy to create a particle with mass equal to the particle you are trying to measure. That level of accuracy is called the compton wavelength of the particle, and its formula is:
Lc = h / mc
Or, in English, "The compton length of a particle is h (Planck's constant) divided by its mass times the speed of light." This is derived from the quantum-mechanical equations describing the behaviour of very small masses.
There is another distance which you can work out for a given mass, its Schwarzschild radius, which is, if you like, the radius of the sphere into which the mass must be compressed in order to form a black hole. The formula for this one is:
Ls = Gm / c2
In English: The Schwarzschild radius is Newton's constant, G, times the mass over the speed of light squared. At this scale the equations of general relativity are crucial in determinining the behaviour of a given mass.
If we then ask for what mass these formulas are equal, ie when:
h / mc = Gm / c2
it's pretty easy to work out that
hc / G = m2
so these two lengths are equal for a mass equal to sqrt(hc/G), which is called the Planck mass. It's about the same mass as a largish cell.
We can now take that mass and calculate what its Schwarzschild radius is (or Compton wavelength, since they are equal.)
This gives us what is called the Planck length, and its formula is:
Lp = sqrt( hG / c3 )
(in English: the square root of hG over c cubed)
which can be calculated as about 1.6 x 10-35 meters, thousands of times smaller to a proton than a proton is to us. Very small indeed!
Information from John Baez's physics web pages, at: