We study a dense colloidal suspension confined between two quasiparallel glass plates as a model system for a supercooled liquid in confined geometries. We directly observe the three-dimensional Brownian motion of the colloidal particles using laser scanning confocal microscopy. The particles form dense layers along the walls, but crystallization is avoided as we use a mixture of two particle sizes. A normally liquid-like sample, when confined, exhibits slower diffusive motion. This appears to be in part because the particle motion is quasi-two-dimensional within the layers. Particle rearrangements are spatially heterogeneous, and the shapes of the rearranging regions become more planar upon confinement. This is related to an increase in the non-Gaussian character of the displacement distributions upon confinement. The wall-induced structure and changing character of the spatially heterogeneous dynamics appear strongly connected with the confinement induced glassiness.