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We show how accretion rate governs the physical properties of a sample of unobscured broad-line, narrow-line, and lineless active galactic nuclei (AGNs). We avoid the systematic errors plaguing previous studies of AGN accretion rate by using accurate accretion luminosities (L_int) from well-sampled multiwavelength SEDs from the Cosmic Evolution Survey (COSMOS), and accurate black hole masses derived from virial scaling relations (for broad-line AGNs) or host-AGN relations (for narrow-line and lineless AGNs). In general, broad emission lines are present only at the highest accretion rates (L_int/L_Edd > 0.01), and these rapidly accreting AGNs are observed as broad-line AGNs or possibly as obscured narrow-line AGNs. Narrow-line and lineless AGNs at lower specific accretion rates (L_int/L_Edd < 0.01) are unobscured and yet lack a broad line region. The disappearance of the broad emission lines is caused by an expanding radiatively inefficient accretion flow (RIAF) at the inner radius of the accretion disk. The presence of the RIAF also drives L_int/L_Edd < 10^-2 narrow-line and lineless AGNs to 10 times higher ratios of radio to optical/UV emission than L_int/L_Edd > 0.01 broad-line AGNs, since the unbound nature of the RIAF means it is easier to form a radio outflow. The IR torus signature also tends to become weaker or disappear from L_int/L_Edd < 0.01 AGNs, although there may be additional mid-IR synchrotron emission associated with the RIAF. Together these results suggest that specific accretion rate is an important physical "axis" of AGN unification, described by a simple model., Accepted for publication in the Astrophysical Journal. 15 pages, 9 figures