To croon is to sing a soft or emotional song. A father might croon a lullaby to his baby as she falls asleep. Elvis Presley was known to croon to the ladies.
The verb croon is originally a Scottish word, from the Middle Dutch root kronen, "to lament or mourn." When you croon, you hum or sing in a soft, low voice, like your heart is breaking. You might croon to your child or sweetheart, or maybe you croon in the shower. In the 1930s and 40s, a "crooner" was a singer who was known for specializing in sentimental jazz songs.