The first post does a good job in explaining the general concept of the Trinity. Being Roman Catholic, I just thought I would throw out Saint Patrick's awesome (and easy to understand) explanation as well.
The Trinity is comprised of God the Father, God the Son (Jesus Christ), and God the Holy Spirit (or the Holy Ghost). Three persons, ... one God. Saint Patrick told the people of Ireland that the best way to understand this Triune God was to look at their very own shamrock. A shamrock, being a little plant that grows all over Ireland, has three leaves protruding from its one stem. Of course, it is still only one plant: a shamrock. Simply said: three leaves, one plant. Simply said: three Persons, one God.
There is one God who has one son. Also, there is one Holy Spirit. These three distinct beings represent one kingdom and make up the Godhead. It can be compared to a royal king who has relatives that make up one family.
As a Christian, one believes that there is one God who has one son and one Holy Spirit.
Jesus Christ, God's only son is just that. Jesus came to earth in flesh form to become the ultimate sacrifice to bridge the gap(which resulted when Adam fell from grace) between God and man. God created man in his own image, but man in his human nature committed error. Jesus Christ and his crucifixion became the redemptive plan for mankind's salvation.
The Holy Spirit came to earth after Jesus Christ was resurrected from the dead and ascended back to the throne to be seated next to God the father.
Now there is God the father, God the son (Jesus Christ) and God the Holy Spirit.
The Holy Spirit lives inside of mankind if he is invited to do so. He acts as a teacher, a comforter and guide.
God the father, God the son and God the Holy spirit are three distinct beings with one purpose in mind and that is to reconnect with mankind after the fall of Adam and Eve.