The Father is the G-d who acts throughout the Old Testament and talks to Christ in the New Testament.
The Son is Jesus, whose sacrifice on the crucifix either ransomed souls from hell or opened the portal to heaven for whoever wants to go, depending on whom you ask. He was both G-d and man, not considered to be some kind of phantasm or soulless possessed being, but was just like other humans except for also being G-d.
The Holy Spirit is the little angel sitting on cartoon characters' shoulders, telling them the proper way to deal with the universe. It is within everyone, the part of G-d that communicates directly with humans.
All three parts are widely held to be coeternal, of the same substance, and yet inexplicably different. All are considered to be present at each stage in history; the Trinity just shows the appropriate character at a given point.