This paper reports the results of experimental studies of the dynamics of particles in a dense quasi-two-dimensional colloidal liquid. We find that at high density, near close packing but still in the liquid phase, the spatial configurations of the particles in the colloidal liquid consist of small ordered domains separated by disordered boundaries. There are frequent exchanges of particles between the ordered and disordered domains, so the lifetime of a particular ordered domain is short and the state of the system is ergodic. The motion of a particle in an ordered domain is constrained but fully two dimensional. The motion of a particle in a disordered boundary has considerable one-dimensional file-server character. By virtue of exchanges of particles between the ordered and disordered domains, the time dependence of the particle displacement has mixed character. We find that the particle dynamics in the dense quasi-two-dimensional colloidal liquid can be characterized with three simultaneous competing relaxation processes, each of which contributes to the mean square displacement of a particle, W(t). For an interval that is shorter than the time between collisions, W(t) is linear in time; this behavior results from particle displacements within a cage of fluctuating neighbors. At somewhat longer time W(t) has a sublinear dependence on time. We associate this temporal behavior with a file-server-like contribution to W(t), arising from correlated motion in the disordered boundary regions. At very long time we find that there are contributions to W(t) from infrequent large displacements, of the order of a particle diameter in length. The large displacements are associated with density fluctuations that fluidize the surroundings of a particle. These large displacements are not ballistic; rather the particle diffuses rapidly enough to generate a large displacement per unit time. The common occurrence of structural heterogeneity in simulation studies of systems with different particle-particle interactions and different equations of motion suggests that fluctuations that generate local order in a dense two-dimensional liquid may be a ubiquitous feature of that class of systems. The results of the experiments reported in this paper suggest that current theories of relaxation processes in a dense quasi-two-dimensional liquid do not adequately take into account the class of fluctuations that generates transient ordered domains in the liquid; nor do these theories adequately account for the cooperative relaxation of those domains.