As an adjective vested means to be absolutely promised as a benefit or legal right. To be vested in a company is to have been given certain benefits or privileges.
It can also mean having a vest, which is a sleeveless garment worn over the torso.
A related phrasal noun is vested interest. Vested interest means an additional motive for desiring a certain action or outcome. This additional motive usually is personal or not relating to the main motive.
Sometimes vested is used as shorthand for having a vested interest. However, this may cause confusion between having a right to something or simply having a desire for that outcome.