The objective of this research is to investigate the target controllability of second-order multi-agent systems (MASs), which is a critical issue in understanding and designing complex networked systems. Through the node partition and matrix decomposition, this paper establishes a necessary condition for the system, and then a full necessary conditions for judging the target controllability is obtained by discussing the left Jordan chain of the system matrix. Furthermore, this paper specifically focuses on second-order MASs based on path graphs, star graphs and complete graphs, and then provides some precise and efficient methods for the target controllability. In order to verify the correctness of theoretical findings, a series of numerical simulations are conducted.