In a homogeneous two-dimensional system at non-zero temperature, although there can be no ordering of infinite range, an ordered superfluid phase is expected to occur for a Bose liquid. Theory predicts that, in this phase, the correlation function decays with distance as a power law, and quantum vortices are bound to antivortices to form molecular-like pairs. We study the relevance of this theory to microcavity exciton polaritons. These are two-dimensional bosonic quasiparticles formed as a superposition of a microcavity photon and a semiconductor quantum well exciton, and have been shown to condense at high enough densities. Because of the short lifetime, equilibrium is not established, but we instead probe the steady state of the system, in which particles are continuously injected from a pumping reservoir. We employ a Michelson interferometer setup to measure the first order spatial correlation function of such a condensate. The gaussian form of the short-distance decay allows us to define an effective thermal de Broglie wavelength, although the system is not in thermal equilibrium. The long-distance decay is measured to be a power law with an exponent in the range 0.9-1.2, larger than is possible in equilibrium. Our non-equilibrium theory suggests that this can be attributed to laser pumping noise. We also present our observation of a single vortex-antivortex pair in a condensate of the appropriate size. Pairs are generated due to pumping noise, and are formed sequentially at the same point due to the inhomogeneous pumping spot profile. They are revealed in the time-integrated phase maps acquired using Michelson interferometry. Our results suggest that vortex-antivortex pairs can be created in a two-dimensional condensate without rotation or stirring. The observed correlated motion of a vortex and antivortex imply that vortex-antivortex pairs do not dissociate, which is consistent with the measured power law decay of the spatial correlation function. These two experiments uniquely describe the condensate phase fluctuations and provide stringent tests to theories of nonequilibrium condensation. They also highlight the exciton polariton condensate as a very well characterized system showing mesoscopic coherence and deepen our understanding of fundamental two-dimensional bosonic physics. Progress in this field is expected to lead towards long-sought applications such as quantum simulation or low-threshold laser sources.