A madrigal is a specific kind of song, one that you won't often hear on the radio. The madrigal developed in 16th-century Italy and is sung in musical counterpoint, by several singers at once.
You're only likely to sing a madrigal if you're in a club or musical group that has an interest in old choral music that is sung a capella (without instruments), with several intertwining voices at once and following a strict poetic form. If you enjoy going to the Renaissance Fair, you'll hear lots of madrigals, because the madrigal form was most popular during the Renaissance. Even though few people sing them today, some of the musical forms and structures from madrigals can be heard in contemporary pop songs.