"The other day" means "a few days ago". You use it to talk about a day that's about 2 to 10 days before today.
So you wouldn't use it if you're talking about something that happened yesterday, and you wouldn't use it for something that happened three weeks ago. You use "the other day" when you don't exactly remember what day something happened, or when it's not important to tell the exact day.
I was talking to somebody about that just the other day.
You can also say "the other night", "the other evening", etc.
"The other day" and "the other night" can be used in business or casual situations.