Most Restaurants in Japan limit everything. For example, restaurants often have a two hour seating limit. Free refills are rare in Japan. Family Restaurants have no time limit. They're the type of place in which you can sit and study for hours without feeling guilty. Coffee and beverages are (usually) unlimited — drinks are serve-yourself from a "drink bar".
As the name suggests, family restaurants are popular with families. Most have both Western and Japanese food items. There's usually something on the menu that appeals to every generation — grandma and grandpa, mom and dad, teenagers and kids.
Food at family restaurants is reasonably high quality (as a rule). Healthy options are available.
Unhealthy options are also available.
Family restaurants are a great place to study. Japanese apartments are often tiny, deary places to study. Family restaurants have comfortable, spacious seating (far better than chain cafes). Seating is often modeled after American diners of the 1950s.
Family restaurants are most popular with teenagers and university students on a budget. Many people spend half their youth chatting away with friends at family restaurants.
Japanese are often nostalgic about family restaurants. Most people have spent countless evenings wasting the night away in their favorite family restaurant.
Big family restaurant chains include Johnathan's (ジョナサン), Denny's (デニーズ ~ different from the US chain) , Saizeriya (サイゼリヤ), Coco's (ココス), Skylark and Royal Host (ロイヤルホスト). In recent years, several chains have closed locations. Some worry family restaurants are fading out in favor of fast food.