A giant star is a star larger than a main sequence star. After the main sequence, stars go through phases where they become giants, e.g., red giant or a star on the asymptotic giant branch.
The name originally applied to especially bright stars, i.e., a large luminosity or absolute magnitude. Because the brightness is directly related to the area facing us (the square of the radius), the brightest stars are indeed large, known as bright giants or for even brighter, supergiants. The general term dwarf star actually applies to any star that is not a giant, sometimes used with a qualifier to signify a more specific meaning (white dwarf, red dwarf, brown dwarf).