Background Imaging of Cosmic Extragalactic Polarization (BICEP) is a bolometric polarimeter designed to measure the inflationary B-mode polarization of the cosmic microwave background (CMB) at degree angular scales. During three seasons of observing at the South Pole (2006 through 2008), BICEP mapped ~2% of the sky chosen to be uniquely clean of polarized foreground emission. Here we present initial results derived from a subset of the data acquired during the first two years. We present maps of temperature, Stokes Q and U, E and B modes, and associated angular power spectra. We demonstrate that the polarization data are self-consistent by performing a series of jackknife tests. We study potential systematic errors in detail and show that they are sub-dominant to the statistical errors. We measure the E-mode angular power spectrum with high precision at 21 < ell < 335, detecting for the first time the peak expected at ell ~ 140. The measured E-mode spectrum is consistent with expectations from a LCDM model, and the B-mode spectrum is consistent with zero. The tensor-to-scalar ratio derived from the B-mode spectrum is r = 0.03+0.31-0.26, or r < 0.72 at 95% confidence, the first meaningful constraint on the inflationary gravitational wave background to come directly from CMB B-mode polarization.