What is a person? One way of approaching the concept of person is to follow a thread from personality to self to person, interweaving notions of narrative (language), society, and physical individuality.
Personality:
A set of dispositions, behaviours, and impressions to which a coherent and distinguishable narrative can be attached (this is not to suggest a requirement of integrity or consistency as the notion of a scattered or incoherent personality is itself a coherent narrative, i.e. a stable and clear judgement).
Personality is not exclusively human. It can be applied to other animate beings and even inanimate objects. It presumes the aforementioned narrative, the type we use when assessing/predicting the state of others onto which we can project characteristics we note in ourselves during the process of self-awareness (in order to understand others in some relation to ourselves and therefore formulate ways to react to or control them) but it doesn’t presume self-awareness.
We can correctly talk of cats and dogs and even places or things as having personalities because we can derive from them a consistency of disposition, behaviour, and impression that equates to the same basic standard of narrative we apply to persons. A human, animal, thing or place may equally (though in different ways) be referred to as, e.g. creepy, unpredictable, and untrustworthy or bright, friendly and happy.
Self:
A self is a personality + conceptual self-awareness. That is, a narrative that continuously refreshes itself through self-observation, a self-sustaining, self-referential narrative, a story that writes itself on the basis of itself. Being a narrative, it is fundamentally linguistic (conceptual) and being linguistic it is fundamentally social. And in order to be the particular narrative we refer to as a “self”, it requires a limit so that it may relate to that which allows its expression. Its social aspect is dependent then on that limit, which is the body. So a self is a self-creating abstraction fuelled by the interaction of the linguistic layered over the social and sustained by the concrete differentiation of bodies. And analagous to words' status as meaningful abstractions being sustained by their differentiation from each other in linguistic systems, we selves are meaningful self-referential abstractions sustained by our concrete differentiation in bodies in social systems. The only selves we know of are located in human bodies.
Person:
A person is a socially named self (self-aware set of behaviours, dispositions, and impressions located in a body). Or how society refers to selves. Recognized formally/legally under the concept of personhood and informally as one of the concept “people”. Another way to put it is "person" is how society conceptualizes its social atoms of meaning, just as words are conceptualized as linguistic atoms of meaning.