For Aristotle, psyche or “soul” is an ordinary empirical concept. First and foremost, it names a set of functional characteristics and abilities that distinguish living from nonliving things. Associations with what we might call specifically mental functions are secondary to this. The Stanford Encyclopedia of Philosophy has a good introduction.
When speaking of “parts” of the soul, Aristotle means distinguishable groups of functions. He explicitly leaves open the question whether they are actually separable from one another. Sometimes he seems to distinguish three main parts: one associated with the basic functions of nutrition and reproduction shared by all living things, including plants; one associated with desire and sensorimotor functions, shared by all animals; and one associated with reason and intellect. At other times, he makes finer distinctions.
Perhaps unexpectedly, reason (logos) and intellect (nous) for Aristotle seem to be more sharply distinguished from what we might call mental functions than mental functions are from biological ones. In contrast with, e.g., memory, which he says cannot exist independent of a living body, and in spite of the fact that he thinks thought contents build on sense perception, he explicitly says intellect has no bodily organ and comes from outside. The very closely connected concept of reason is mainly associated with language and the right kind of socialization. I think reason and intellect still get treated as part of the soul mainly because their presence reshapes the whole (See also Reasonableness; Feeling; Second Nature; Intelligence from Outside; Psyche, Subjectivity; Passive Synthesis, Active Sense.)
For animals, including rational animals, Aristotle located the seat of the soul in the heart. In his time, there was as yet no evidence to refute this traditional conception; understanding of the physiological role of the brain and nervous system only developed in later Greek medicine. But recall also that the psyche for Aristotle is at root more vital than mental. Even today, we still distinguish life by a literal heartbeat, and associate emotion figuratively with the heart.