More greetings probably exist for the Yuletide than any other holiday. One can say, "Merry Christmas" or "Happy Christmas" or "Happy Holidays" or any number of other ways. The variety of greetings for Christmas most likely exist because of the antiquity of the holiday and its importance to Northern Europe and later North America as well.
Sources from the era do not preserve the holiday greetings which may have existed in Old English. In fact, the word Christmas is not even preserved in Old English as we know it. It occurs only as a compound, Cristes mæsse "Christ's mass." More often than not, Christmas was simply referred to as Geol, our modern word Yule. Of course, Geol was originally the name of the pre-Christian, midwinter celebration observed by the ancient Germanic peoples, from which Christmas may take many of its trappings (the Yule log, holly, eating ham for the holiday, et. al.). It was called in Old Norse by essentially the same name--Jól. If we are to look for what Christmas greetings were in Old English, then perhaps we should look to the other Germanic languages for a clue. Looking at Danish, Norwegian, and Icelandic, the greeting is very nearly the same. It is Glædelig Jul in Danish, Gledelig Jul in Norwegian, and Gleðileg Jól in Old Icelandic. In Old Norse, then, the typical Yuletide greeting may have been Glaðligr Jól. Glaðligr was an Old Norse word meaning, "happy, cheerful, bright;" it is a cognate to our modern English word glad. In Old English, then, the typical Yuletide greeting may have simply been Glæd Geol--literally, "Glad Yule."
Of course, "Merry Christmas" is the most common holiday greeting in the United States and Canada, and is still heard frequently in the United Kingdom as well. Despite this, the phrase "Merry Christmas" is not terribly old. Although the word merry goes all the way back to Old English, the words merry and Christmas do not appear to have been used in combination until the 16th century, when it appears in the traditional carol "We Wish You a Merry Christmas." Its first use in prose would not be until 1699, in an informal letter written by Admiral Francis Hosier of Deptford, England. The phrase would not catch on until the 19th century, however, when it became very popular. It was used in the very first Christmas card in 1843 and also appears in A Christmas Carol by Charles Dickens, first published that same year. Late in the 19th century the phrase "Happy Christmas," which appears in "A Visit from St. Nicholas" from 1923, would make a bit of comeback in the United Kingdom. It is for this reason that while "Merry Christmas" seems to be the dominant phrase in the United States, in the United Kingdom one will hear "Happy Christmas" nearly as often, if not more so.
In person English speakers might wish each other "Merry Christmas" or "Happy Christmas," but if they send a card it might well say, "Season's Greetings." Perhaps to avoid repetition, Victorian Christmas cards bore a variety of greetings, among them "Christmas Greetings," "Yuletide Greetings," and "Compliments of the Season." Eventually the phrase "Seasons greetings" developed late in the 19th century as a typical holiday greeting on Christmas cards.
While "Seasons Greetings" is typically reserved to Christmas cards, the phrase "Happy Holidays" is one with which people might greet each other. Contrary to what some might think, "Happy Holidays" is not a phrase which developed recently. I am not sure when the phrase "Happy Holidays" originated, but it was in common usage by the 20th century. By 1940 the phrase was common enough that it was used for the title of a Yuletide themed Columbia animated short. And, of course, "Happy Holiday" is an Irving Berlin song from the classic Holiday Inn. Indeed, at least by the mid-20th century the phrase "Happy Holidays" was being used on Christmas cards. I am not sure why or how the phrase "Happy Holidays" developed, but it may well have been due to a desire for a greeting that would include both Christmas and New Year's Day. Of course, today it has become a common greeting in various stores and other services, who might not always know the religious persuasion of their customers. This has led to controversy in some quarters, with certain individuals concocting a whole "War on Christmas."
While holiday greetings have changed over the years and have been the source of controversy among some, they will always exist in some form as long as the holiday does. In fact, holiday greetings would appear to be an important part of the season. They are a means of individuals, even total strangers, sharing the holiday with each other.