Exciton-polaritons are hybrid quasi-particles, eigenstates of the strong coupling regime between light and matter. Their Bose-Einstein condensation has been recently demonstrated, opening the way to several fundamental investigations on the associated phenomenology and topological defects. Major advantages are brought by the unique light-matter mixed nature of polaritons, allowing for example for condensation at high temperatures and direct optical access to the condensate wavefunction. This thesis is devoted to the investigation of polariton condensates, with particular attention to the role of disorder, which represents an intrinsic and pronounced feature of the CdTe-based microcavity samples under study. The experiments are carried on in several configurations and conditions for the condensates, with particular attention to coherence behavior and topological defect formation and stabilization. One-dimensional polariton condensates have been studied in linear disorder valleys where, despite the potential fluctuations, the non-equilibrium processes underlying condensation allow for the phase-locking of single-energy condensates. Even in such conditions, disorder is responsible for the observed pronounced fragmentation of the condensate density, leading to a non-trivial modulation of the spatial coherence. The investigations have been carried further, performing a full quantum tomography of one-dimensional polariton condensates. The reconstruction of the single-particle density matrix of the system allowed to directly test and validate, in a strongly out-of-equilibrium framework, the Penrose-Onsager criterion for Bose-Einstein condensation, which was initially formulated for equilibrium systems. By shaping the non-resonant laser excitation, one-dimensional polariton condensates could be tamed to occur also within the annular geometry. Periodic boundary conditions are there imposed by the closed ring topology. In such setting, upon spontaneous condensation, further ordering occurs in the condensate density in the form of standing-wave patterns. Theoretical insight into the pattern formation are brought by Gross-Pitaevskii modeling, reproducing reasonably well the experiments. The pump shape anisotropy together with disorder are responsible for the pinning into a preferential orientation of the spatial pattern, allowing the observation in our time-integrated experiments. Two-dimensional polariton condensates have also been extensively investigated. The attention was focused onto topological defects formation, dynamics and stabilization within disorder in the spinor condensate framework. Polaritons indeed constitute a two-component spinor condensate. Previous works brought experimental observation of both integer and fractional vorticity, spatially pinned in the disorder potential. Yet, in our system, the formation dynamics of such entities and the inter-relation of integer and fractional vorticity was to be experimentally explored. Ultrafast spectroscopy experiments have been designed and performed, allowing for the study of the dynamics of spinor topological defects. In particular, the dissociation of integer vortices into pairs of fractional vortices was demonstrated. A novel disorder-dependent coupling acting on the spin components of the condensate has been introduced in order to reproduce the observed phenomenology within a generalized spinor Gross-Pitaevskii model. In two-component spinor polariton condensates, integer spin vorticity has also been predicted to occur under specific configurations, yielding formation of spin textures into the condensate polarization structure. Advanced interferometric techniques with polarization sensitivity have been extensively employed to eventually provide the first observation of such entities in polariton fluids. They are spontaneously forming and pinning into disorder potential traps in the sample. The polarization textures have been also directly imaged and their stabilization properties understood in terms of the realistic polarization features of the microcavity sample. The generalized spinor Gross-Pitaevskii model is once more successfully employed to catch the physics of spin vortices formation and stabilization within the landscape of a realistic disorder.