“Hi Jane, I'm not feeling well this morning.”
You don't want to go to work this morning because you stayed out late last night drinking and now you have a hangover. But you don't want to tell your boss that that's the reason, so you write an email to her starting with this.
Hi Jane, I'm not feeling well this morning.
In most jobs in the U.S., this is an appropriate way to start an email to your boss. You can use your boss's first name, and it's appropriate to say "Hi" as the greeting. It would seem quite strange to write to your boss in a really formal way like this:
Dear Ms. Harris,
I'm not feeling well this morning.
"I'm not feeling well" is a good set phrase you can use when you want to tell someone that you feel sick, but you don't want to specifically say what's wrong. "I'm not feeling well" can be used when you have a headache, stomachache, a cold, diarrhea, menstrual cramps, or other health problems. You probably wouldn't use it if you broke your leg, had a heart attack, or had another major health problem.
It's common to say "I'm not feeling well" as an excuse for not going to work. Your boss might believe you, or might not. But most bosses won't ask you more specifically why you need to stay home because that information is considered to be very personal.