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You must always capitalise Fathers' Day when you're referring to the official holiday.
When you're writing Fathers' Day, you must always use an apostrophe after Fathers (before Day), as this holiday belongs to every father in the world, not just one. When you're writing about something that more than one person owns — for example, students' grades are getting worse every year — you must, in most cases, use an apostrophe after the group's name (after the s); for example, weed is teens/chavs' favourite thing to smoke. However, if you're writing something that only one person owns — for example, Dwayne's a gay chav, and he's proud to admit it — you must use an apostrophe after the person's name (before the s).
Remember to write your Fathers' Day cards correctly, people. This will likely help your dads understand them.