Naturalness is a concept specific to
effective field theory (EFT). An EFT is a description of the physics at energy scales below some cutoff \Lambda.
A simple illustration of an EFT is the following. Let's suppose that at high energies, we have a massless particle \phi and a particle X with mass M, along with some interaction between them. To be even more specific, let's consider these as scalar fields with a certain quartic interaction, so that the system is described by a Lagrangian
L = \frac{1}{2} (\partial \phi)^2 + \frac{1}{2} (\partial X)^2 + \frac{M}{2} X^2 + g \phi^2 X^2. (*)
This is a nice, renormalizable QFT.
Now, effective field theory comes in when we study the system at energies E< M. At these low energies, we can create states of arbitrary numbers of \phi particles, but the energy is not large enough to generate an X particle, since it is too massive. Nevertheless, quantum corrections certainly involve X particles in intermediate states. So the effective description at low energies should just involve the \phi particle, with a Lagrangian that has new terms representing the interactions between \phi particles given by exchanging X particles in loops. Such a Lagrangian can be written as
L_e = \frac{1}{2} (\partial \phi)^2 + c_2 \phi^2 + c_4 \phi^4 + c_6 \phi^6 + \cdots . (**)
The resulting effective field theory is not renormalizable because of the infinite number of terms involved, but it can still give meaningful results for processes occurring at energy scales sufficiently lower than the cutoff scale \Lambda = M.
Now naturalness is just the statement that if we were to use the Lagrangian (*) to compute the coefficients c_i in (**), we would find that
c_i = \alpha_i \Lambda^{4-i},
where the \alpha_i are of "order one." That is, the \alpha_i would take values around 0.01-100, rather than 10^{-9} or 10^6.
Naturalness is a bias for what "looks right," it's not a provable criterion. It's more the statement that, if one of the coefficients is of order one, then there's no reason for the others not to be of order one. Along with this is the notion that if a coefficient is extremely close to zero, then this is only natural if there is a new symmetry that is restored in the limit that the coefficient is exactly zero.
The model above is too simple to illustrate any of these symmetries, but it can illustrate the hierarchy problem in the Higgs sector of the Standard Model. In that case, we can consider (**) to be a model for the Higgs potential and \Lambda the scale of new physics beyond the Standard Model. There are various possibilities, such as supersymmetry, a GUT, or just gravity. The Higgs mass and quartic coupling are roughly
m_H \sim \alpha_2 \Lambda^2,~~~ \lambda_H \sim \alpha_4 .
The quartic coupling is indeed of order one, so that \alpha_4 is natural. However, since m_H \sim 10^2~\mathrm{GeV}, \alpha_2 is unnaturally small if \Lambda\sim 10^{16}~\mathrm{GeV}, as in the case where the only new physics beyond the SM is a GUT. The situation is even worse if there is no new physics until the Planck scale.
Supersymmetry is a popular solution to the hierarchy problem. Besides the control over corrections he to the Higgs mass, it also makes the Higgs mass natural, since \Lambda \sim 10^3~\mathrm{GeV}, so that \alpha_2\sim 0.1 or so, which is natural enough.
As for other examples of naturalness, one can also look at the effective field theory describing the Standard Model after the electroweak symmetry is broken, so \Lambda \sim 250~\mathrm{GeV}. In particular, the Yukawa couplings for the quarks are interesting to look at. Since the top quark mass is \sim 175~\mathrm{GeV}, this is an example of a natural coupling. The bottom and charm quark masses are also natural, but the up, down and strange quarks are tending toward unnaturalness. However, when these quark masses are zero, SU(3) flavor symmetry is restored, so this isn't viewed as a big naturalness problem.
The lepton sector is a bit more interesting, since the electron mass is fairly unnaturally small (though to a much smaller extent than the Higgs mass problem). This would be less of a problem if there was some sort of family symmetry that was restored in a GUT.
