Who doesn't know this typically German word which doesn't have a real equivalent in English. Maybe you could call it cosy but it doesn't really explain the word. When a restaurant is "gemütlich", it means that there is a nice temperature - not too hot or cold-, you are comfortably seated, there is a nice atmosphere and you feel at home.
Specific to German restaurants - I know it also from Austria and Switzerland - is the fact that you may come to eat there and then stay there all evening until you decide to go home. It isn't expected of you that you change for a bar or other location after you have finished your meal.
We even have a song which describes it
In this case it is a song which is frequently sung at beer festivals.
If you like other German drinking songs, try out this one: