We present new radio observations of the ''prototypical'' ultraluminous far-infrared galaxy NGC 6240, obtained using the VLA at lambda = 20 cm in B-configuration and at lambda = 3.6 cm in A-configuration. These data, along with those from four previous VLA observations, are used to perform a comprehensive study of the radio emission from NGC 6240. Approximately 70% (similar to 3 X 10(23) W Hz(-1)) of the total radio power at 20 cm originates from the nuclear region (less than or similar to 1.5 kpc), of which half is emitted by two unresolved (R less than or similar to 36 pc) cores and half by a diffuse component. The radio spectrum of the nuclear emission is relatively flat (alpha approximate to 0.6; S-nu) proportional to nu(-alpha). The supernova rate required to power the diffuse component is consistent with that predicted by the stellar evolution models of Rieke et al. (1985). If the radio emission from the two compact cores is powered by supernova remnants, then either the remnants overlap and form hot bubbles in the cores, or they are very young (less than or similar to 100 yr). Nearly all of the remaining 30% of the total radio power comes from an ''armlike'' region extending westward from the nuclear region. The western arm emission has a steep spectrum (alpha approximate to 1.0), suggestive of aging effects from synchrotron or inverse-Compton losses, and is not correlated with starlight; we suggest that it is synchrotron emission from a shell of material driven by a galactic superwind. Inverse Compton scattering of far-infrared photons in the radio sources is expected to produce an X-ray flux of similar to 2- 6 x 10(-14) ergs s(-1) cm(-2) in the 2-10 keV band. No significant radio emission is detected from or near the possible ultramassive ''dark core'' hypothesized by Bland-Hawthorn, Wilson, & Tully (1991).