Good question! I hope many people chime in here as this varies depending on where you're from.
In British variants "holiday" is used as a more general term so I'll let someone else explain its usage! If you're interested in learning how we use it in the US please read on!
In the US holiday usually refers to specific dates which are commonly celebrated. Many times these are actually designated by our federal or local government. Some examples in the US are Independence Day (July 4th), Memorial Day, and Christmas. When someone request time off from work we typically call that a "vacation". Vacations can also have include the assumption that you are traveling. I could have a long weekend because of a holiday and take a vacation during that time.
(Additionally, different companies will use different terms for vacation days/time off. Luckily, this does not really have an effect on daily conversation.)
Personally, I might say:
I want to book some time off.
I requested some vacation time.
I filled out the time off request form.
My boss approved my request for time off.
I've used up my vacation days.
I get a new allotment of vacation days in April.
I get vacation days in a year.
(My company uses the term PTO (paid time off) so in reality I just say the letters P-T-O if it's related to my work schedule.)
I hope that was helpful!