Carve-Out
Humans naturally want to carve out a place in society (think: caves, clubs or the comment sections on YouTube). But companies also like to carve things out. In their case, this concept means to offload a business unit to form a new company.
Yes, known as a carve-out, there are two kinds: the mother company does not sell the business altogether, but rather sells a minority interest in it to outside investors or business partners. Think: the cable industry competitors each owning, say, 12% of CNN. In the second common form, the mother company divests itself of the business unit while making that unit a standalone company. Think: Google selling off its Boston Robotics division after it didn't click with the founders.
In the spin-off scenario, rather than selling shares to anyone, current stockholders are given shares in the new company. The parent might still retain some equity for old times' sake. Usually the business unit being carved out is not part of their core operations, such as when a cosmetic company carves out a division that produces food products.
Companies carve out divisions because they might be looking to raise cash, but they still want to maintain some strategic business connection with the new company and/or have some control over decision making. Many times a carve-out can result in a higher stock price for the parent company if investors think the company would be better off without that business unit or because investors "pay up" for a pure play investment rather than have a small but exciting business buried within a behemoth conglomerate.