We introduce a model for diffusion of two classes of particles (A and B) with priority: where both species are present in the same site the motion of A 's takes precedence over that of B 's. This describes realistic situations in wireless and communication networks. In regular lattices the diffusion of the two species is normal, but the B particles are significantly slower due to the presence of the A particles. From the fraction of sites where the B particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where B 's are free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes, leads to trapping of the low priority particles in scale-free networks.