This paper presents an adaptive fault-tolerant control (FTC) scheme for a class of nonlinear uncertain second-order multi-agent systems. A local FTC component is designed for each agent in the distributed system by using local measurements and suitable information exchanged between neighboring agents. Each local FTC component consists of a fault diagnosis module and a reconfigurable controller module comprised of a baseline controller and two adaptive fault-tolerant controllers activated after fault detection and after fault isolation, respectively. Under suitable assumptions, the closed-loop stability and leader-follower formation properties of the distributed system are rigorously established under different operating modes of the FTC system, including the time-period before possible fault detection, between fault detection and possible isolation, and after fault isolation.