  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 (M_{0.8}) above an
extinction threshold of A_K approximately equal to 0.8 magnitudes,
corresponding to a gas surface density threshold of approximaely 116 solar
masses per square pc. We argue that this surface density threshold corresponds
to a gas volume density threshold which we estimate to be n(H_2) approximately
equal to 10^4\cc. Specifically we find SFR (solar masses per yr) = 4.6 +/- 2.6
x 10^{-8} M_{0.8} (solar masses) 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 star formation rates in molecular clouds and
galaxies is to understand those physical factors which give rise to the dense
components of these clouds.
