In this paper, we consider the leader-follower synchronization problem for a class of second order nonlinear multi-agent systems with unknown dynamics and environmental disturbances. The proposed scheme utilizes only local relative state information, it is fully distributed and solves the problem with transient and steady state performance guarantees. In fact, the designed controller has build-in the capability of enforcing pre-determined and designer specified performance bounds on the state tracking errors. Despite the high amount of uncertainty, the controller is structurally and computationally simple, avoiding the use of approximating structures i.e., neural networks, fuzzy systems etc. Simulations are also provided to clarify and verify the approach.