What is the physical significance of Compton wavelength? Wikipedia says,

The Compton wavelength represents the quantum response of mass to local geometry.

What does it actually mean?
 A: Another, better way, (because imo, the wiki line you quote is a bit obscure)  of dealing with  your question is based on  John Baez's Notes:

The Compton wavelength of a particle, roughly speaking, is the length scale at which relativistic quantum field theory becomes crucial for its accurate description.
A simple way to think of it is this. Trying to localize an electron to within less than its Compton wavelength makes its momentum so uncertain that it can have an energy large enough to make an extra electron-positron pair! This is the length scale at which quantum field theory, which describes particle creation, becomes REALLY important for describing electrons. The Compton wavelength of the electron is the characteristic length scale of QED (quantum electrodynamics).
It's easy to guess how big the Compton wavelength is using the knowledge that it depends only on the mass of the electron, relativity and quantum mechanics. Mass has dimension M. Length has dimension L. Time has dimension T. In relativity we have a constant, the speed of light, with dimensions L/T, and in quantum mechanics we have a constant, Planck's constant, with dimensions ML2/T = energy times time = momentum times position. These two constants enable us to express units of mass in terms of dimensions of inverse length. In this case, the Compton wavelength is about 4 × 10$^{-13}$ meters.
In fact, this is usually called the "reduced" Compton wavelength. What people usually call the Compton wavelength is 2π times as big, about 2 × 10$^{-12}$ meters. That's because the wavelength of a wave is really not the reciprocal of its frequency: it's 2π divided by the frequency.

