What is Fantastic Romance? Well, to put it simply, it's a Fantasy subgenre that has a fantasy backdrop but has more emphasis on the characters' romantic interactions. It is a combination of two of the world's most popular genres: Fantasy and Romance.
Now, this subgenre goes way back to the medieval European era and the Renaissance period. More people began to learn about literature during these times and combined literary fiction with earlier myths and legends, which then created the Fantasy subgenres. And Fantastic Romance was one of the first that appeared.
Romance was already a well-known genre and Fantasy was a perfect match. Together, they not only followed traditional myths, legends, fables, etc., but added new fantastical elements with the love story. During the Renaissance, the trend with Romance began to lean towards fantastic fiction. A great example would be The Faerie Queene by Edmund Spenser. It's a poem, notably one of the longest poems in the English language, but it has romantic and fantastical features. It was a typical knightly romance with duels and combats against sorcerers, giants, and goblins.
A common plot or theme, which you may have already encountered, in this genre, would be a group of adolescents who may have been orphans or outcasts, who are drawn together through destiny, only to discover they possess special abilities. Their destiny might be saving a kingdom from an evil villain, but along the way, they fall in love. One example similar to this could be the Red Queen series by Victoria Aveyard, stories in which romance and revolution collide.
Settings usually vary. The characters could be in present time, living in a city. Or they could be in the 1800's and trying to survive through the Seven Years' War. Magic exists in any time period or place and so does love.
Characters in Fantastic Romance are generally young and lost in the world. Throughout the story, they grow to be strong and usually discover they come from a magical background. Either that or they fall in love with a magical or supernatural being and encounter fantastic issues they have to overcome. A fairly popular example of this would be the Twilight series by Stephenie Meyer, where Bella Swan falls in love with Edward Cullen who is a vampire.