The whole family got together __ Christmas.
Meaning: Christmas is the reason for the family getting together
The word usage is wholly dependent upon what you are trying to say with the sentence. Note that the differences, while real, are also minor. Most readers would guess the reason for the gathering.
This implies that the family got together around Christmas time, but does not suggest that Christmas was the reason for the gathering. This can be seen in the following example:
If you are trying to say the family gathered to celebrate the holidays, then you might say
It depends on what you're trying to say (as noted by others).
"On": refers to that day: "We got together on Christmas for dinner and a gift exchange." "We went to a movie on Christmas because everything else was closed."
"For": refers to purpose: "We got together for Christmas" implies that Christmas was the reason (versus the movie example above).
"Over": refers to a period of time including that day; "He went home from college for a week over Christmas."
I see that others have suggested "at", which would not have occurred to me (mid-western US English).
As Christmas describes a period of time, I think that during is the most fit:
The whole family got together during Christmas.
I would say "at" is correct, but "for" would also work. It's not a preposition, but "to celebrate" is a bit more descriptive (although in a lot of cases probably untrue:) )
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