Trinity

Christians believe that there is only one God, and that He is the Author and Director of the Universe. He is awesome, magnificent and beyond comprehension.

His wisdom is profound, his power is vast.
  Who has resisted him and come out unscathed?
He moves mountains without their knowing it
  and overturns them in his anger.
He shakes the earth from its place
  and makes its pillars tremble.
He speaks to the sun and it does not shine;
  he seals off the light of the stars.
He alone stretches out the heavens
  and treads on the waves of the sea.
He is the Maker of the Bear and Orion,
  the Pleiades and the constellations of the south.
He performs wonders that cannot be fathomed,
  miracles that cannot be counted. (Job 9:4-9)

Yet this one God should actually be thought of as having three "persons" within the God-head.

"When we profess to believe in one God, by the name God is understood the one simple essence, comprehending three persons or hypostases." (John Calvin)

Someone tried to explain this by using this plant, which is called a shamrock.

There is only one shamrock in the picture but there are three distinct parts to it. The same can be said of God; there is one God but three distinct "persons."

The three "persons" are called the Father, the Son, and the Holy Spirit.

Father

"To the Father is attributed the beginning of action, the fountain and source of all things; to the Son, wisdom, counsel, and arrangement in action, while the energy and efficacy of action is assigned to the Spirit." (John Calvin)

Son

About 2000 years ago the Son took on human nature and became known to the world as Jesus Christ.

The Lord of the universe took the form of a servant; the impassible God became a suffering man; the Immortal One submitted himself to the dominion of death; Majesty assumed into itself lowliness; Strength, weakness; Eternity, mortality. (Leo I)

The result of the incarnation, that infinite act of divine love, is the God-Man. Not a double being, with two persons; nor a compound middle being; but one person, who is both divine and human. Christ has a rational human soul, and a human will, and is therefore in the full sense of the word the Son of man; while yet at the same time he is the eternal Son of God in one person, with one undivided self-consciousness. (Philip Schaff)

Holy Spirit

The third person of the Trinity is known as the Holy Spirit.

The Holy Ghost is nowhere in Scripture reckoned among creatures or angels, but is placed in God himself, co-eternal with God, as that which searches the depths of Godhead (1 Cor. ii. 11, 12). He is the source of all gifts in the church (1 Cor. xii). He dwells in believers, like the Father and the Son, and makes them partakers of the divine life. The Holy Ghost is the source of sanctification, and unites us with the divine life, and thus must himself be divine. (Philip Schaff)

Conclusion

The doctrine of the Trinity is one which no man will ever understand in its entirety.

The Church fathers did not pretend to have exhausted the mystery of the Trinity, and very well understood that all human knowledge, especially in this deepest, central dogma, proves itself but fragmentary. All speculation on divine things ends in a mystery, and reaches an inexplicable residue, before which the thinking mind must bow in humble devotion. (Philip Schaff)

However there are some general points which we should remember

There is only one divine essence or substance. Father, Son, and Spirit are one in essence, or consubstantial. They are in one another, inseparable, and cannot be conceived without each other. In this point the Nicene doctrine is thoroughly monotheistic or monarchian, in distinction from tritheism, which is but a new form of the polytheism of the pagans.

In this one divine essence there are three persons or, to use a better term, hypostases, that is, three different modes of subsistence of the one same undivided and indivisible whole, which in the Scriptures are called the Father, the Son, and the Holy Ghost.

These distinctions are not merely different attributes, powers, or activities of the Godhead, still less merely subjective aspects under which it presents itself to the human mind; but each person expresses the whole fulness of the divine being with all its attributes, and the three persons stand in a relation of mutual knowledge and love.

The Father communicates his very life to the Son, and the Spirit is the bond of union and communion between the two. The Son speaks, and as the God-Man, even prays, to the Father, thus standing over against him as a first person towards a second; and calls the Holy Ghost "another Comforter" whom he will send from the Father, thus speaking of him as of a third person.

In each person there is the same inseparable divine substance, united with the individual property and relation which distinguishes that person from the others. The word person is in reality only a make-shift, in the absence of a more adequate term. Our idea of God is more true and deep than our terminology, and the essence and character of God far transcends our highest ideas. (Philip Schaff)