"The other day" means "a few days ago". You use it to talk about a day that was 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.
Here's one example that you can use when writing to someone you met last week: