set up or lay the groundwork for
"establish a new department"
establish the validity of something, as by an example, explanation or experiment
"The experiment demonstrated the instability of the compound"
"The mathematician showed the validity of the conjecture"
To make stable or firm; to confirm.
"The strong foundation helped establish the new tower against the fierce winds, confirming its stability for everyone watching."
In plain English: To establish something means to set it up so that it exists and is accepted as real or official.
"The new company plans to establish its headquarters in the city next year."
Usage: Use establish to mean setting up an organization, custom, or fact and making it secure or widely accepted. Avoid using it simply to indicate that something already exists or is currently happening.
The word "establish" comes from the Middle English establissen, which was borrowed from Old French. It ultimately traces back to the Latin stabilire, meaning "to make firm or steady."