Thomas Hobbes is famous for saying that in a state of nature, everyone would be at war with everyone else, "and the life of man, solitary, poore, nasty, brutish, and short" (p. 186). Hobbes did not really think that this state of nature had ever happened in the past. It was instead what he believed would happen if there were no government, and more or less what happened in a civil war, or when government completely broke down.
Thus, Hobbes believes that government is necessary to keep from living in constant danger, struggle, and violence. But how does government come about? Hobbes suggests it comes about by a sort of contract or agreement that people make, to let someone (or some group of people) have complete authority over all of them. To explain this, he talks about contracts, or what he calls covenants.
In everyday speaking, we use "person" to mean any individual human being. But "person" also has another meaning, which is what Hobbes uses. "A person is he whose words or actions are considered, either as his own, or as representing the words or actions of an other man, or of any other thing to whom they are attributed, whether Truly or by Fiction" (p. 217). A "person," then, is someone who can say or do things. He may speak or act for himself or herself, or for someone else.
Speaking or acting for someone else is what Hobbes means by "representation." A lawyer may represent me in court, speaking to the judge and jury on my behalf. An actor might represent me in a play, speaking and acting as though he were me. In either case, the lawyer or actor is "bearing my person." As Hobbes puts it:
So that a Person, is the same that an Actor is, both on the Stage and in common Conversation; and to Personate, is to Act, or Represent himselfe, or an other; and he that acteth another, is said to bear his Person, or act in his name ... and is called in diverse occasions, diversly, as a Representer, or Representa...