The 23rd Grammy Awards were held in 1981, and were broadcast live on American television. They recognized accomplishments by musicians from the year 1980.
However, the Grammys, usually held in February, (last of what are considered the "big three" music awards shows, including the BMA and AMA shows) are considered the approximate equivalent to the Oscars, in the music world.
Unlike the Academy Awards, for which the eligiblilty period begins January 1, the eligibility period for the Grammys begins October 1, which results in September being considered the Christmas sales period for the music industry (in which artists generally release big albums to qualify for the next year's Grammy).