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Under II. "BODY AND SOUL BUT TRULY ONE", from the Catechism of the Catholic Church, 365, the Church teaches that:
365 The unity of soul and body is so profound that one has to consider the soul to be the "form" of the body:1 i.e., it is because of its spiritual soul that the body made of matter becomes a living, human body; spirit and matter, in man, are not two natures united, but rather their union forms a single nature.
Therefore according to Catholic teaching, man's one nature is the union of spirit and matter, soul and body.
The section mentioned above continues
367 Sometimes the soul is distinguished from the spirit: St. Paul for instance prays that God may sanctify his people "wholly", with "spirit and soul and body" kept sound and blameless at the Lord's coming.2 The Church teaches that this distinction does not introduce a duality into the soul.3 "Spirit" signifies that from creation man is ordered to a supernatural end and that his soul can gratuitously be raised beyond all it deserves to communion with God.4
Therefore the Church teaches that while the soul is sometimes distinguished from the spirit, the distinction does not introduce a duality into the soul.
1. Cf. Council of Vienne (1312): DS 902.
2. 1 Thess 5:23.
3. Cf. Council of Constantinople IV (870): DS 657.
4. Cf. Vatican Council I, Dei Filius: DS 3005; GS 22 § 5; Humani Generis: DS 3891.
Further reading:
The section The soul in Christian thought in the article Soul | New advent.
Addendum after reading @PeterTurner's answer.
In the New Testament the word spirit and, perhaps, even the expression spirit of God signify at times the soul or man himself, inasmuch as he is under the influence of God and aspires to things above; more frequently, especially in St. Paul, they signify God acting in man[.] - Holy Ghost | New Advent.
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If the "or" in the question is exclusive, it is a loaded question, as spirit "spirit" (spiritus) and "soul" (anima) are really two aspects of a human soul, which is one and undivided:
As St. Thomas Aquinas writes in Summa Theologica I q. 97 a. 3 c.:
…rational soul is both soul and spirit. It is called a soul by reason of what it possesses in common with other souls—that is, as giving life to the body…But the soul is called a spirit according to what properly belongs to itself, and not to other souls, as possessing an intellectual immaterial power.
Also, man is made in the image of God because of his intellectual soul. St. Augustine says in Gen. ad lit. vi, 12 (cf. Summa Theologica I q. 93 a. 2 sed contra):
Man's excellence consists in the fact that God made him to His own image by giving him an intellectual soul, which raises him above the beasts of the field.
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One addendum to the highly referenced answer from FMS I'd like to make is that at Mass in the several times in which we say "and with your spirit" we are talking about the spirit in which the priest was ordained.
Therefore sometimes when we talk about a person's spirit what we're really talking about is the in-dwelling of the Holy Spirit.
I think you might also refer to it as The Kingdom of God within you.