'I'm on holiday' is the most common and standard way, and you might also hear 'I'm on my holidays.' 'I'm on holidays' is incorrect.

Meaning to go away during a break, then "on vacation" is the term in the US and "on holiday" is the term in the UK. "The Holidays" or "The Holiday Season" is also used in the US to refer specifically to the break for most businesses and schools that occurs in winter, that usually includes the holidays Hanukkah, Christmas, Kwanzaa, the Winter Solstice, and New Years Day (and possibly more). You will probably hear the term "Heading home for The Holidays" which refers to many people in the US traveling during this time of year to visit their parents or grandparents, and sometimes other relatives.

"On vacation" is used in the US, "on holiday" is used in the UK and "on holidays" is used in Australia.