A public library: a library run by a public authority (city, ward, etc.) that is open for public use, lending books and providing reading/research services.
この町には公共図書館が二つある。
There are two public libraries in this town.
公共図書館では図書カードを作れば本を借りられる。
At the public library you can borrow books if you make a library card.
夜でも勉強できるように、公共図書館は開館時間を延長した。
So people can study in the evenings, the public library extended its opening hours.