The word "establish" is similar to the words create, build, make, and form. We use "establish" as a verb when describing how, why, where, or when something started:
According to this sign, this business was established in 1905. (est. 1905)
This group was established in 1967 to help prevent the destruction of this wilderness area. (Notice the use of the passive voice in this example.)
Maria is trying to establish a good credit history by paying her loans and credit cards on time.
Tuan has established himself as a respectable member of his community.
The police were brought in to establish order in an area that had known only chaos for two months.
Bill and Melinda want to establish a treatment facility for children who have chronic diseases.
The word "established" is an adjective. It usually means that a person or thing is recognized as reputable or as having good credentials.
Stephen Johnson is an established member of the community.
An established author is visting our local bookstore this afternoon.
David decided to buy an established business instead of starting one from scratch.
To make the noun form of this word, just add "ment" to the end of it:
The tavern that we usually go to is a very old establishment. (Sometimes the word "establishment" is used in place of the word "business," particularly when the business is associated with a building.)
The establishment of a government took three years.
The U.S. Constitution prohibits the establishment of an official religion in the United States.
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances. (From the First Amendment to the U.S. Constitution.)