The ##\overline{\mathrm{MS}}## scheme is defined within dimensional regularization, which is very elegant from the calculational point of view, but it's not very intuitive.
As the most simple example take simple ##\phi^4## theory with the Lagrangian
$$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \phi)(\partial^{\mu} \phi)-\frac{m^2}{2} \phi^2 -\frac{\lambda}{4!} \phi^4,$$
which is renormalizable. The divergent parts are (besides the here not so interesting vacuum diagrams) the self-energy (leading to wave-function and mass renormalization), and the four-point function (leading to coupling-constant renormalization).
The counterterms and the renormalization scheme are thus determined by these divergent pieces. One can write the counter-term Lagrangian in the form
$$\delta \mathcal{L} = \delta Z \frac{1}{2} (\partial_{\mu} \phi)(\partial^{\mu} \phi) - \frac{1}{2} (\delta m^2 + m^2 \delta Z_m) \phi^2 - \frac{\delta \lambda}{4!} \phi^4.$$
Here the ##\delta Z##, ##\delta Z_m## and ##\delta \lambda## are all dimensionless (i.e., of energy dimension 0). Only ##\delta m^2## has dimension 2.
The most intuitive MIR scheme is defined by introducing a mass scale ##M## as the renormalization scale and define the renormalized quantities by the renormalization conditions
$$\Sigma(p^2=0,m^2=0)=0 \; \Rightarrow \; \delta m^2=0.$$
This is allowed, because that's a quadratically divergent quantity and thus IR safe. So you can define it at ##m^2=0##.
All other divergences are logarithmic and thus cannot be defined at ##m^2=0## and all external momenta at 0. Thus one defines them at ##m^2=M^2##, i.e.,
$$[\partial_{p^2} \Sigma(p^2,m^2)]_{p^2=0,m^2=M^2}=0 \; \Rightarrow \; \delta Z,$$
$$[\partial_{m^2} \Sigma(p^2,m^2)]_{p^2=0,m^2=M^2}=0 \; \Rightarrow \; \delta Z_m,$$
$$\Gamma^{(4)}(s=t=u=0,m^2=M^2)=-\lambda \; \Rightarrow \; \delta \lambda.$$
All the counter terms are thus only dependent on the renormalization scale ##M^2## via the dimensionless renormalized coupling constant ##\lambda##, but not on ##m^2##. That's why this scheme is a mass-independent renormalization scheme, and you have ##\delta Z=\delta Z(\lambda)##, ##\delta Z_{m}=\delta Z_m(\lambda)##. The RG parameters like ##\beta## don't depend on ##m/M## but only on ##M## through ##\lambda##, and you get a homogeneous RG equation that is not so difficult to solve.
For details, see my QFT manuscript
http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf
Chpt. 5.
The derivation of the ##\beta## function and other RG equation coefficients within the MIR, MS (or ##\overline{\text{MS}}##) scheme in the context of dim. reg. see Sect. 5.11. In Sect. 5.12 also other ("non-MIR") schemes are treated. The corresponding RG equations are more difficult since the RG coefficients do not only depend on the renormalization scale ##M## via the dimensionless renormalized coupling ##\lambda## but also explicitly via ##m/M##.