Relatively early in the research of nanotubes, Thess et al. calculated the resistivity of ropes of metallic SWNTs to be in the order of 1E-4 ohm-cm at 300 K. They did this by measuring the resistivity directly with a four-point technique. One of their values they measured was 0.34E-4 ohm-cm, which they noted would indicate that the ropes were the most highly conductive carbon fibers known, even factoring in their error in measurement. In the same study his measurements of the conductivity, Frank et al. was able to have reach a current density in the tube greater than 1E7 A/sq cm. Later, Phaedon Avouris suggested that stable current densities of nanotubes could be pushed as high as 1E13 A/cm2.A SWNT is a Single Walled Nano Tubes.
So how does that compare to copper? For household wiring typical current density is 500A/sq cm and ultimate current density is maybe 10X that with the wires near the melting point or beyond. In round numbers 1E4 A/sq cm vs 1E7 A/sq cm for carbon nanotubes. In other words 1,000 times the current density. At a weight per unit volume of about 1/4 that of copper. Copper resistivity at room temperature is about 1.7E-4 ohm-cm. So carbon nanotubes can carry about 5X as much current as an equivalent volume of copper for the same losses.
If we can get this stuff into mass production - which is likely to take twenty or thirty years - we can rewire the grid we have for 5X times as much power as it handles now or the same power with 1/5th the losses. Not room temperature superconductors, but a definite improvement.
Update: A good overall look at Carbon Nanotubes: Carbon nanotubes (CNTs): A small review
H/T IntLibber at Talk Polywell