This paper addresses the leader-following formation problem of nonholonomic mobile robots. In the formation, only the pose (i.e., the position and direction angle) of the leader robot can be obtained by the follower. First, the leader-following formation is transformed into special trajectory tracking. And then, a neural network (NN) finite-time observer of the follower robot is designed to estimate the dynamics of the leader robot. Finally, finite-time formation control laws are developed for the follower robot to track the leader robot in the desired separation and bearing in finite time. The effectiveness of the proposed NN finite-time observer and the formation control laws are illustrated by both qualitative analysis and simulation results.