The original campaigner for creating Mother's day, Anna Jarvis, explicitly wanted the apostrophe to be before the "s".
Take any group of people and ask them this question. Does Mother's Day need an apostrophe, and — if so — where does it go?
After the fighting stops, the combatants will have settled into three camps. Let's look at all three in turn.
Incidentally, because the names of holidays are written with an initial capital, we write Mother's Day, not Mothers day, regardless of where you put the apostrophe, if any.
Choice 1. Mothers Day: no apostrophe
The argument here is that Mothers do not own the day, so no possession is involved. No apostrophe is thus needed. We are describing a day for Mothers, not a day belonging to Mothers.
Choice 2. Mother's Day: an apostrophe before the s
Here the argument is that the day belongs to one specific Mother (yours presumably). So, because possession is involved, Mother's Day needs an apostrophe.
Choice 3. Mothers' Day: an apostrophe after the s
Here the argument is that the day is shared among all Mothers collectively. We thus need an apostrophe after the s.
And the winner is...
As shown above, you can make a reasonable case for all three of the choices. This article makes clear, though, that the original campaigner for creating Mother's day, Anna Jarvis, explicitly wanted an apostrophe, and she wanted it to be before the "s":
... it was to be a singular possessive, for each family to honour their mother, not a plural possessive commemorating all mothers in the world.
Having said this, you will continue to see all three forms.
If, following Anna Jarvis's wishes, you employ the usage "Mother's Day", it would be consistent to write "Father's Day" in the same manner.