Molly Malone's is an old-fashioned Irish Pub. It feels like a home away from home. The service is fast and friendly, the food is good, and the music at night is very good. Try to go on a night with an Irish band. There's even an Irish bartender on staff during lunches.
If you are in Tallinn alone and looking to meet up with some friends or have a little romance, I recommend Molly's. You won't feel uncomfortable entering and having a drink alone, and you probably won't stay alone very long. If you're an attactive woman, you will be approached, but you won't be harrassed or bothered if you make it known that you're not looking for conversation.
Try to make it in during a soccer match. They televise the matches and the scene at the bar is very fun.
Even though this is a bar, it's very much family place during the day and early evenings. It's a great place for the family to stop for lunch or dinner.