The formal way to say "Happy birthday" in Spanish is "Feliz cumpleaños," but most native Spanish speakers will shorten it to simply "Cumpleaños" or "Felicidades" which means congratulations.
English speakers do not generally congratulate people on their birthdays, but in most Spanish-speaking countries, it is customary and correct to use that in place of "happy birthday." The phrase "¡Felicidades en tu día!" is another birthday greeting and translates to "Congratulations on your day!" It is important to note that the word "tu" is the familiar form. For people that are not known well, an individual should use the more formal "su" in place of "tu" for the word "your."