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Copulas provide an attractive approach for constructing multivariate densities with flexible marginal distributions and different forms of dependence. Of particular importance in many areas is the possibility of explicitly modeling tail-dependence. Most of the available approaches estimate tail-dependence and correlations via nuisance parameters, yielding results that are neither tractable nor interpretable for practitioners. We propose a general Bayesian approach for directly modeling tail-dependence and correlations as explicit functions of covariates. Our method allows for variable selection among the covariates in the marginal models and in the copula parameters. Posterior inference is carried out using a novel and efficient MCMC simulation method.