で (de) is one of the most frequent particles in real use and has two central values: the place where an action happens and the means by which it is done. Do not confuse it with に, which with existence verbs marks where something simply is.
As the place of action: 図書館で勉強する (としょかんでべんきょうする / toshokan de benkyō suru) “I study at the library”. Compare に, used for static existence: 図書館に本がある (としょかんにほんがある / toshokan ni hon ga aru) “there is a book in the library”. The difference is between an action taking place somewhere (で) and something being located somewhere (に).
As means or instrument: 電車で行く (でんしゃでいく / densha de iku) “I go by train”, 箸で食べる (はしでたべる / hashi de taberu) “I eat with chopsticks”. で also covers material (木で作る (きでつくる / ki de tsukuru) “made of wood”), cause (病気で休む (びょうきでやすむ / byōki de yasumu) “be absent because of illness”), and scope (“within, in total”).