We present ultraviolet, optical, and near-infrared observations of SN 2012ap, a broad-lined Type Ic supernova in the galaxy NGC 1729 that produced a relativistic and rapidly decelerating outflow without a gamma-ray burst signature. Photometry and spectroscopy follow the flux evolution from -13 to +272 days past the B-band maximum of -17.4 +/- 0.5 mag. The spectra are dominated by Fe II, O I, and Ca II absorption lines at ejecta velocities of 20,000 km/s that change slowly over time. Other spectral absorption lines are consistent with contributions from photospheric He I, and hydrogen may also be present at higher velocities (> 27,000 km/s). We use these observations to estimate explosion properties and derive a total ejecta mass of 2.7 Msolar, a kinetic energy of 1.0x10^{52} erg, and a 56Ni mass of 0.1-0.2 Msolar. Nebular spectra (t > 200d) exhibit an asymmetric double-peaked [OI] 6300,6364 emission profile that we associate with absorption in the supernova interior, although toroidal ejecta geometry is an alternative explanation. SN 2012ap joins SN 2009bb as another exceptional supernova that shows evidence for a central engine (e.g., black-hole accretion or magnetar) capable of launching a non-negligible portion of ejecta to relativistic velocities without a coincident gamma-ray burst detection. Defining attributes of their progenitor systems may be related to notable properties including above-average environmental metallicities of Z > Zsolar, moderate to high levels of host-galaxy extinction (E(B-V) > 0.4 mag), detection of high-velocity helium at early epochs, and a high relative flux ratio of [Ca II]/[O I] > 1 at nebular epochs. These events support the notion that jet activity at various energy scales may be present in a wide range of supernovae.
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