Does Roman Catholicism teach that man is body and soul (bipartite) or body, soul, and spirit (tripartite)? According to Rev. S. D. F. Salmond's English translation, John of Damascus wrote,(1)


  For since man’s nature is twofold, consisting of soul and body, He bestowed on us a twofold purification, of water and of the Spirit: the Spirit renewing that part in us which is after His image and likeness, and the water by the grace of the Spirit cleansing the body from sin and delivering it from corruption, the water indeed expressing the image of death, but the Spirit affording the earnest of life. 


Does Roman Catholicism teach that a human consists of body and soul, or body, soul, and spirit?



References

(1) Exact Exposition of the Orthodox Faith, Book IV, Chapter IX.
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.