This paper considers a Markovian model of a service system motivated by communication and information services. The system has finite processing capacity and offers multiple grades of service. The highest priority users receive a "guaranteed" processing rate, while lower priority users share residual capacity according to their priority level and therefore may experience service degradation; hence the term "best effort." This paper focuses on performance analysis for this class of systems. We consider the Halfin-Whitt heavy-traffic regime where the arrival rate and system processing capacity both grow large in a way that the traffic intensity approaches one. We first derive a multi-dimensional diffusion approximation for the system dynamics, and sub-sequently obtain a more tractable diffusion limit based on an intuitive "perturbation approach." This method enables us to compute various closed form approximations to steady-state as well as transient congestion-related performance measures. Numerical examples illustrate the accuracy of these approximations.