Romance is simply the wonderful relationship that takes place between a man and a woman whose hearts have been attracted to each other. Perhaps the book in the Word of God which best describes a godly romantic relationship is The Song of Solomon.
Remember too that romance should never end after marriage begins! The romantic love that blossomed before marriage ought to develop and grow all during that union. "Rejoice with the wife of thy youth...be thou always ravished with her love" (Prov. 5:18,19). "Husbands love your wives" (Eph. 5:25, Col. 3:19).
Courtship is a formal term used in English speaking lands which describes the actions that take place in a romantic relationship. A striking example of courtship is seen in Gen. 29:18-20. Jacob loved (romance) Rachel and agreed to serve her father Laban for seven years (courtship) to have her as his wife. We don't think of courting in our Western society as work—but that was what it meant in the culture in which Jacob and Rachel lived!
Dating is really just another, more commonly used word for courtship. We might also say that going on a date is one of the activities of a courtship. Too, we often say that when a man and woman are attracted to each other and begin to "court", they are "dating" or "going steady".