We derive convenient uniform concentration bounds and finite sample multivariate normal approximation results for quadratic forms, then describe some applications involving variance components estimation in linear random-effects models. Random-effects models and variance components estimation are classical topics in statistics, with a corresponding well-established asymptotic theory. However, our finite sample results for quadratic forms provide additional flexibility for easily analyzing random-effects models in non-standard settings, which are becoming more important in modern applications (e.g. genomics). For instance, in addition to deriving novel non-asymptotic bounds for variance components estimators in classical linear random-effects models, we provide a concentration bound for variance components estimators in linear models with correlated random-effects. Our general concentration bound is a uniform version of the Hanson-Wright inequality. The main normal approximation result in the paper is derived using Reinert and Röllin's (2009) embedding technique and multivariate Stein's method with exchangeable pairs.