Biblically speaking, nothing.
The Bible doesn't teach human beings are a "trinity", or what is more commonly call tripartism or trichotomism. Nor has this ever been the historic Christian perspective.
More common has been what is sometimes called dichotomism, man is a dichotomy of his material self (the body) and his immaterial self (the soul or spirit).
Sometimes this dichotomism also becomes a problem, biblically and theologically, when it gets forced too hard into an overtly Platonic dualism; Platonic dualism would view "the soul" as the "true self" and the body as in some sense lesser, inferior, or a facade hiding the true self of the soul. Think of what Yoda says in Star Wars, "Luminous beings are we, not this crude matter" this is a very Platonic way of seeing things, and taken excessively leads to various Gnostic and heretical ideas--such as that the body is evil and false, because physical matter is inferior/evil.
But taken simply and non dualistically, the dichotomy of body and soul is a helpful way of understanding biblical teaching that human beings were created as rational, thinking, relational, and moral creatures-we aren't just our bodies, but we aren't flighty souls trapped in bodies either. We are fully integrated, fully bodied, fully "souled" creatures who reason, who were created to love God, reflect God, love each other, and be God's Image bearers in creation. This means that when the body dies, we don't merely cease to exist--there is something about us that can exist even without and apart from the body--but this isn't how we were made or intended to exist, it is an unnatural severing of body and soul caused by the unnatural reality of death, which is an affliction upon the whole cosmos wrought by and perpetuating sin.
The Bible, when speaking of human beings, uses a lot of terms to describe the "parts" of human beings--body, mind, soul, spirit, heart--to name some of the most common. We wouldn't, therefore speak of a quadrinity of human beings by speaking also of the mind or the heart; or a quintity by speaking of the mind and heart along with body, soul, spirit. Instead we understand that, biblicaly speaking, and according to the historic faith of the Church, that the human person can be described as "more than the body" and so there is something about us that gives us an "us-ness" even apart from the body (and yet, this body of ours is also very much us--my body is also me); so this dichotomous language that there is a material and immaterial dimension to the human person is something we have historically and continuously seen throughout Christian history, going back to Scripture itself.
There is, therefore, no reason to regard the human soul and the human spirit fully independent things--but rather both "soul" and "spirit" are ways of speaking of the rational, emotive, relational, spiritual, and trans-material quality of the human creature.
-CryptoLutheran
