“I think I'm going to take the day off.”
You want to take the day off work today because you went out drinking last night and you don't feel good this morning. You're writing an e-mail to your boss to let her know that you're not coming in to work. You've explained that you don't feel well, so now you write this.
I think I'm going to take the day off.
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This is how to tell someone what you want to do, if you don't think they will mind.
Hey, I think I'm going to order take-out for dinner.
In a lot of American workplaces, you can use this to ask your boss for permission to take vacation days or sick days. You might expect that it would be more polite to ask in this way:
Can I take the day off?
This isn't wrong, but it does sound a bit like a child asking for something from his parents. If you are allowed to take days off of work and it doesn't cause extra problems for other people, it's better to act a little more confident and tell your boss your plan, rather than asking.
This means to stay home from work. You usually don't use this phrase when you're going on a vacation. Instead, you use it to describe staying home from work because you're sick, because you want to rest, because you have a dentist's appointment, and so on.
If you know how long you're going to take off, you can say "take the day off", "take a couple days off", etc. If you don't know how long it will be, you can say "take some time off":
I've been taking some time off.