で

で marks the place where an action happens, the means by which it happens, or the cause behind it. Compare: 学校で勉強する ("study at school" — action's location) vs 学校にいる ("be at school" — existence's location). Means: 電車で行く — "go by train." Cause: 病気で休む — "rest because of illness."

The action-vs-existence distinction (で vs に) is the single most useful test for choosing between them with locations. If the verb describes an event, motion, or activity happening in that place, use で. If the verb describes the noun being in that place, use に.

Another common use: で also marks the material or composition — 紙で作る ("make out of paper"). And the limiting boundary — 一週間で終わる ("finish in one week"). The unifying intuition is "the frame within which / by means of which X happens."