1. Where does the word "Trinity" comes from?

It comes from the Latin word trinitas, which means "three" or "triad." The Greek equivalent is triados.

2. When was it first used?

The first surviving use of the term (there may have been earlier uses that are now lost)  was around A.D. 170 by Theophilus of Antioch, who wrote:

In like manner also the three days which were before the luminaries, are types of the Trinity [Τριάδος], of God, and His Word, and His wisdom.

 And the fourth is the type of man, who needs light, that so there may be God, the Word, wisdom, man [To Autolycus 2:15].

3. What is the Trinity?

The Compendium of the Catechism of the Catholic Church explains it this way:

The Church expresses her trinitarian faith by professing a belief in the oneness of God in whom there are three Persons:

 Father, Son, and Holy Spirit.

The three divine Persons are only one God because each of them equally possesses the fullness of the one and indivisible divine nature.

They are really distinct from each other by reason of the relations which place them in correspondence to each other.

The Father generates the Son; the Son is generated by the Father; the Holy Spirit proceeds from the Father and the Son [CCCC 48].

4. Is the Trinity the central mystery of the Christian Faith?

Yes. The Compendium explains:

The central mystery of Christian faith and life is the mystery of the Most Blessed Trinity.

Christians are baptized in the name of the Father and of the Son and of the Holy Spirit [CCCC 44].

5. When did the Church infallibly define the Trinity?

The dogma of the Trinity was defined in two stages, at the First Council of Nicaea (A.D. 325) and the First Council of Constantinope (A.D. 381).

First Nicaea defined the divinity of the Son and wrote the part of the Creed that deals with the Son.

This council was called to deal with the heresy known as Arianism, which claimed that the Son was a supernatural being but not God.

First Constantinople defined the divinity of the Holy Spirit and wrote the part of the Creed that deals with the Spirit.

This council dealt with a heresy known as Macedonianism (because its advocates were from Macedonia) which denied the divinity of the Holy Spirit.

 This heresy was also called Pneumatomachianism (from a Greek phrase meaning "fighting the Spirit").