Full rank means no redundancy among the columns.
Why it matters for regression. A rank-deficient design matrix means no unique least-squares solution, because some predictor is an exact linear combination of others. Near-deficiency is multicollinearity, where the solution exists but is wildly unstable.
In covariance estimation: with n assets and only T < n observations, the sample covariance matrix has rank at most T and is therefore singular - it claims some portfolios have exactly zero risk, which is obviously false and dangerous if used for optimisation.