The whole family got together __ Christmas.
Meaning: Christmas is the reason for the family getting together
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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.
The whole family got together at Christmas.
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:
The whole family got together at Christmas for Uncle Benny's funeral.
If you are trying to say the family gathered to celebrate the holidays, then you might say
The whole family got together for Christmas.
The whole family got together to celebrate Christmas.
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).