I would define a carol as either a traditional song on a religious subject, such as I Saw Three Ships Come Sailing In, or a composed song dealing with a religious subject in a traditionally folkloric fashion, such as Tchaikovsky's Crown of Roses or the Huron Carol by Jean de Brebeuf.

They usually focus on a feast, Christmas most of all but also Easter and Pentecost (Whitsun), becuase they are intended for secular use -- enjoyment in a social setting. If you would feel, um, normal singing it in the pub with a beer in your hand, it's a carol. If it seems more natural to sing it in church, it's a hymn.