What does it mean by active quarks?

Without more context, the usual meaning of active quark is quarks whose rest mass is lower than the energy scale of the environment. So for example, the active quarks of $T = 0$ QCD are $u,d,s$ because they all have rest mass lower than the QCD scale, $Lambda_text{QCD}approx 200text{ MeV}$.

In response to your updated question, active quarks are on shell. Basically, you pair up all quarks into diquark quasiparticles, and then any unpaired quarks are the active quarks. Since quarks are identical particles, the specific one you choose to be "active" is arbitrary, but it will necessarily be of a fixed flavour.