The gesture you describe is indeed the Sign of the Cross. Making that gesture is called "crossing oneself".
The Sign of the Cross is called a sacramental. These are "sacred signs which bear a resemblance to the sacraments. They signify effects, particularly of a spiritual nature, which are obtained through the intercession of the Church." [CCC 1667]
When crossing oneself, one is blessing (or, strictly, invoking God's blessing) on oneself. A blessing is not reserved to clergy, laypeople can bless; and particularly can bless themselves.
Sacramentals derive from the baptismal priesthood: every baptized person is called to be a "blessing," and to bless. Hence lay people may preside at certain blessings...
CCC 1671 deals specifically with the Sign of the Cross.
Among sacramentals, blessings (of persons, meals, objects, and places) come first. Every blessing praises God and prays for his gifts. In Christ, Christians are blessed by God the Father "with every spiritual blessing."175 This is why the Church imparts blessings by invoking the name of Jesus, usually while making the holy sign of the cross of Christ.
The sign is not representative of the Trinity per se, even though it normally accompanies the Trinitarian formula "In the name of the Father and of the Son and of the Holy Spirit" (with each of the capitalised words of the Persons coinciding with the ends of the arms of the cross in the gesture). It represents the Cross, the symbol of Christ's triumph over death.
CCC 617 says
His most holy Passion on the wood of the cross merited justification for us.450
and thus we use that symbol of his passion as the symbol of justification and the sign of blessing.
175: Cf Ephesians 1:3
450: Council of Trent: DS 1529.