We introduce an online network formation game: starting with a base graph and a set of candidate edges, at each round, player one picks an edge and reveals it to player two, then player two decides whether to accept it; player two can accept a limited number of edges and makes online decisions aiming to achieve optimal properties (e.g., the number of spanning trees, algebraic connectivity, and total effective resistance) in the synthesized network. Online network formation arises in cooperative multiagent systems, such as robots establishing a secure network in a changing uncertain environment, or individuals forming teams in social networks. We propose a primal-dual algorithm framework for this problem. At each round the algorithm updates the dual solution using all information from previous rounds, and decides the weight on the new edge based on the complementary slackness conditions. We give interpretations of the algorithm for different graph objectives, and derive a bound on the competitive ratio of the algorithm for the log-determinant problem.