The most common Romance languages are Spanish, Portuguese, French, Italian, Romanian and Catalan. Other lesser-known ones include Occitan, Galician, Asturian, Sicilian, Corsican and Sardinian. In all there are about 35 living Romance languages. Sadly their name comes from the word Roman and not the word romantic.
The Romance languages are the ones that evolve from vulgar Latin. Vulgar here means common, ie the version of Latin actually spoken by the common people, compared to the classical Latin of the church and of the elite. Originally vulgar Latin and classical Latin were mutually intelligible, but over time vulgar Latin evolved into the various Romance languages and the people were no longer able to understand classical Latin.
Most Romance languages have lost some of the more difficult aspects of Latin, such as declensions and cases. Because of this they have a much stricter subject-verb-object word order, and they make more use of prepositions.
There are about 800 million speakers of Romance languages in the world and most of them are in Europe, Africa and the Americas. In Europe the places where Romance languages are spoken roughly equates to the boundaries of the Western Roman empire.
And to return to the statement in the first paragraph about the Romance coming from Roman and not romantic, there is a link between the words. Back in those days, “serious” literature was written in classical Latin. Popular tales, such as love stories, were written in the common (ie Romance) language, and so they came to be called romances.