An assignee is the person that gains a right after something is transferred to them legally, such as property or other trust rights.
In real estate terms, the person that is buying the property is considered the assignee, as they will gain the rights contained in the document that are essentially assigned to them by the previous owner. This would include the right to register their name in the property records and the right over the lands and assets on the property.
An assignee has legal rights and requirements that they must follow. Once the assignee gains possession of the asset, they will be legally responsible for any requirements that the asset brought with it. An example of this would be taxes on property.