1784, Josiah Brown, Great Britain. Parliament. House of Lords., Reports of cases, upon appeals and writs of error, in the high court of Parliament, from the year 1701 to the year 1779. Volume 11, page 444:
...the present Earl had no colour or pretence of right to the estate; and yet the appellants had thought fit to buy him out, ...
2006, Percy Harris, Forty Years in and Out of Parliament, page 18:
... and when he complained met with the reply that if he did not like his partner's methods he could either clear out or buy him out.
To close someone's contract by paying him or her a sum of money, the terms of which are often stated in the contract itself.
1986, Punch Imlach, Heaven and Hell in the NHL, page 149:
A standard NHL contract, agreed to by both players and owners, has what is called a buy-out clause. If a player's performance deteriorates to the point that he is put on waivers and there are no takers, meaning nobody in the NHL wants him, his club can buy him out at one-third of what's left on the contract.
When referring to a property, the purchaser often desires to obtain complete control of a piece of real estate, or to purchase a neighboring property, often with the intent to develop the property by building a new or larger building(s), particularly with more housing units.
When referring to a share of a partnership or company, the purchaser is often buying the share of a spouse, other family member, or retiring shareholder.
When referring to a contract, the contract is typically for high ranking executives of a company, professional sports players, professional models, etc.