We have determined detailed radio spectra for 26 compact sources in the starburst nucleus of M82, between 74 and 1.3 cm. Seventeen show low-frequency turnovers. One other has a thermal emission spectrum, and we identify it as an H II, region. The low-frequency turnovers result from absorption by interstellar thermal gas in M82. New information on the active galactic nucleus candidate 44.01+595 shows it to have a nonthermal falling power-law spectrum at the highest frequencies and that it is strongly absorbed below 2 GHz. We derive large magnetic fields in the supernova remnants of order (1-2)(1 + k)(2/7)phi(-2/7) mG; hence, large pressures in the sources suggest that the brightest ones are either expanding or are strongly confined by a dense interstellar medium. From the largest source in our sample we derive a supernova rate of 0.016 yr(-1).