There is one God who is manifested as three persons. We insist on emphasizing both the oneness of God and the distinction between the three persons that God is, and insist further that there is no "better" way to look at it. Somehow, one God is three persons, and all three persons are fully God--all of this makes the Trinity a great mystery. It is thus not supposed to be entirely comprehensible, though things can be said about it that render the idea more tangible. I'll try to say some such things. Christians believe in a personal God--a God that can relate to his creations in a way that is somehow like how we relate to each other. In fact, how we relate to each other is supposed to be an image for what God is like, since we were created in his image. The point of describing God as three persons is not to suggest that there are three distinct Gods, but rather that God is "personal" in three distinct and essential ways. One way to think of it is as follows: if you've ever heard the phrase "God is love," (which is asserted in the bible) notice that, since love describes a relationship, for God to be love he must be at least two persons. The first person in the Trinity, the Father, is the lover, and he loves the second person in the Trinity, his beloved Son. What this is saying is that God is capable of both giving and receiving love: this "exchange" is essential to what God is. The Holy Spirit, who is the third person in the Trinity, can be understood as the love itself--what Christianity is saying is that this love relationship between the Father and Son is just as much a person as the lover or the beloved--all three are personal, and all three are essential for God to be what he is.
In humanity, relationships of love are meant to follow this model, since we are made in the image of God. Take for example, a family: the husband is the lover, the wife is the beloved--both are persons, both are necessary for the family to be what it is. Further, the love between the husband and wife produces (provided all goes smoothly ) another person, their child. This is only an image, though. God made humanity both as an image of this relationship and so that we could really participate in this relationship of love that He is. In terms of one's personal relationship with God, Christians come to know God the Father through Christ, who is the Son, and by doing so God the Holy Spirit dwells within them. In turn we love each other as Christ loves us, as this whole process becomes a participation in the amazing love that God is.
The "love" model of the Trinity doesn't tell the whole story though. It only describes the essence.