Time off means a break or a vacation (from work or school)

I'm going to take some time off of work, so I can go hiking.
I need some time off or I'm going to go crazy! 

I personally don't use the expression "off time." Off time means that you have a regular, busy, schedule, with few breaks. Down time is used more in my opinion than "off time." Actors often use this term to mean the time between seasons of a show, or between films. 

In my down time (off time), I go to my beach house in Malibu.
In his off time (down time), he likes to pay golf. 

"Day off" is a day where you aren't working/are not busy. These can be regular times that you aren't busy, or they 

I'm taking time off, so I can have a day off. 
Tuesdays are my day off. 

An "off day" usually means that something is wrong. This comes from the expression, "something is off," meaning something is wrong or not going well (something that usually goes well or is important).

I don't know why, but for some reason, I'm having a really off day today.
Friday was a really off day for me. 

If you have any more questions, I'd be happy to help you. 

Greg