Download the Shazam application for your smart phone. Start the program and hold your phone up to the speaker. Shazam will "listen" to the lyrics and tell you who sings the song. Shazam also will recognize the song if you sing the lyrics into the phone yourself.
Visit a song lyrics website such as SongLyrics.com or eLyrics.net. Both sites allow you to search for song lyrics by song title or by words in the lyrics. So, if you know that they were "standing on a corner in Winslow, Arizona," you can plug that into the search engine and find out it is the Eagles who sings "Take It Easy."
Go old school. If the Internet and a fancy new phone app can't tell you who sings the song, contact the radio station on which you heard it. Ask for the program director. If she doesn't know the answer, she can look it up on the station's log, which the FCC requires every station to keep.