In this paper, we investigate the level of star formation activity within nearby molecular clouds. We employ a uniform set of infrared extinction maps to provide accurate assessments of cloud mass and structure and compare these with inventories of young stellar objects within the clouds. We present evidence indicating that both the yield and rate of star formation can vary considerably in local clouds, independent of their mass and size. We find that the surface density structure of such clouds appears to be important in controlling both these factors. In particular, we find that the star formation rate (SFR) in molecular clouds is linearly proportional to the cloud mass (M0.8) above an extinction threshold of AK~ 0.8 mag, corresponding to a gas surface density threshold of Σgas~ 116 M☉pc2. We argue that this surface density threshold corresponds to a gas volume density threshold which we estimate to be n(H2) ~ 104/cm3. Specifically, we find SFR (M☉/yr) = 4.6±2.6x10–8 M0.8 (M☉) for the clouds in our sample. This relation between the rate of star formation and the amount of dense gas in molecular clouds appears to be in excellent agreement with previous observations of both galactic and extragalactic star-forming activity. It is likely the underlying physical relationship or empirical law that most directly connects star formation activity with interstellar gas over many spatial scales within and between individual galaxies. These results suggest that the key to obtaining a predictive understanding of the SFRs in molecular clouds and galaxies is to understand those physical factors which give rise to the dense components of these clouds.