Plasma railguns are electromagnetic accelerators used to produce controlled high velocity plasma jets. This thesis discusses the design and characterization of a small coaxial plasma railgun intended to accelerate argon-helium plasma jets. The railgun will be used for the study of plasma shocks in jet collisions. The railgun is mounted on a KF-40 vacuum port and operated using a 90 kA, 11 kV LC pulse forming network. Existing knowledge of coaxial railgun plasma instabilities and material interactions at vacuum and plasma interfaces are applied to the design. The design of individual gun components is detailed. Jet velocity and density are characterized by analyzing diagnostic data collected from a Rogowski coil, interferometer, and photodiode. Peak line-integrated electron number densities of approximately 8 × 10¹⁵ cm^-2 and jet velocities of tens of km/s are inferred from the data recorded from ten experimental pulses.