Consider some deep inelastic scattering $y f\to y f$, where $f$ is a parton inside a nucleon $N$, and $y$ is some particle. The cross-section is then $$ \sigma_{\text{DIS}} = \int dx f_{N/f}(x,Q^{2})\sigma_{yf \to yf}(x,Q^{2}), $$ where $Q$ is called factorization scale.
My very basic question is:
When the parton is plucked from the nucleon it proceeds through some evolution (described by, within the premise of collinear factorisation, DGLAP resummation) up to a so called factorisation scale at which the hard scattering takes place/scale at which PDFs are evaluated. It is analogous to the degree of freedom one has in choosing the renormalisation scale, the scale at which the coupling constant is evaluated at or the scale at which we subtract the divergences from our theory.
In this sense, the factorisation scale $Q^2$ defines the point at which we make a separation of soft and hard dynamics. All low, non-perturbative scales $q^2< Q^2$ are encoded in the PDF while all high scales $q^2> Q^2$ are present in the hard scattering matrix element, susceptible to description via perturbation theory.
For multiple scale processes, the 'best' choice of factorisation scale is not obvious. It is typically chosen so as to minimise the number of large logs arising from particular phase space configurations. Variation of this quantity is a measure of the theoretical uncertainty and can be studied as a reflection of the convergence of the NLO relative to LO contribution etc..
Emissions from the evolution of the incident parton can be considered as part of next to leading order matrix elements or absorbed into lower order PDFs through an appropriate choice of factorisation scale.
