Polology of Feynman amplitudes, Section 10.2, Weinberg In Weinberg's QFT Volume 1 section 10.2, we basically find that we have a pole when we have an intermediate particle state; the momentum of the intermediate state is on-shell. I am having trouble understanding this intuitively. Does this mean that one of the internal lines is an on-shell detectable particle now? Aren't internal lines always "virtual", off-shell particles?
 A: In that section, it is when the particular combination of external momenta $q=q_1+q_2+\cdots+q_r=-q_{r+1}-\cdots-q_n$ goes "on-shell", which in this case means $q^2\to -m^2$ for some physical mass in the spectrum of the theory. This combination of external momenta $q$ can be loosely thought of as the momentum of the exchanged internal particle of mass $m$ via momentum conservation. But since these are external momenta, they are definite values (they are not integrated over) that you can toggle at will to find possible poles.
A: It is the four vector that carries the meaning of "off shell" , by the invariant mass, if its length is  different than the particle named.
The Standard Model uses quantum field theory in four dimensions, with the four vectors of Lorentz special relativity. Thus in the SM mass is not a conserved quantity. What is defined is the invariant mass, the length of the four vector , which is invariant under Lorentz transformations.
Virtual particles in the SM carry the quantum numbers of their name and their propagator has the named particle mass, but are off shell because their mass varies  in the calculation of crossections and decays of the particular interaction calculated, described by the Feynman diagrams.
