To study transport properties of scale-free and Erdos-Rényi networks, we analyze the conductance G between two arbitrarily chosen nodes of random scale-free networks with degree distribution P(k)-k(-lambda) in which all links have unit resistance. We predict a broad range of values of G, with a power-law tail distribution phi(SF)(G)-G(-g(G)), where g(G)=2lambda-1, and confirm our predictions by simulations. The power-law tail in phi(SF)(G) leads to large values of G, signaling better transport in scale-free networks compared to Erdos-Rényi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical "transport backbone" picture we show that the conductances of scale-free and Erdos-Rényi networks are well approximated by ck(A)k(B)/(k(A)+k(B)) for any pair of nodes A and B with degrees k(A) and k(B), where c emerges as the main parameter characterizing network transport.