To institute something is to establish it or set it up, but an institute is an organizational body. Both definitions come from the Latin word instituere, which meant "to put in place". The way they came about differed, however: the verb is through institutus and the noun from institutum, which meant "ordinance". It wasn't too much of a stretch from "established law" to "established organization", and here we are today with the two meanings. Back to the etymology of instituere - we can remove the prefix in-, meaning "in", giving us statuere, which could be interpreted as "to establish" or "to set up". Statuere is from status, meaning "position" (because positions are established; this is also the etymon of English status), and that's from Proto-Indo-European sta, "to stand".
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The Etymology Nerd