Greeting,
it is perhaps easier to define an exclusive process : all particles in the final state are measured/detected/known. The prototype of inclusive process is deep inelastic scattering, for which one measures only the recoiling lepton off a nucleon, and sums over all possible hadronic final states.
Note that there are semi-inclusive processes, whereupon one will measure in addition another. For instance in semi-inclusive DIS, in addition to the lepton one detects another recoiling particle, such as (say for instance) a pion.
Finally, let me note that in exclusive processes, one may not actually detect one of the particles, but constraint kinematically the other particles in such a manner to ensure the only possible undetected recoil to be known.