Well,
Modalism teaches that God is one being who has revealed himself to mankind in three different roles, or modes. In the Old testament God revealed himself as the Father, the almighty, but remote and harsh creator God.
In the gospels, He revealed himself as the Son, the personal, close and human role. As well as the sacrifice etc.
In the present he has revealed himself as the Holy Spirit, God that functions within each of us as comforting and empowerment and revelation.
Thus God the Father, Son, and Holy Spirit are all the same person, just playing different roles.
In Trinitarianism God is one being which is manifested in three eternally co-existing persons
In this view God the Father, son, and Holy Spirit, have all existed together for all time. It is extremely difficult to come up with analogies or word-pictures of the trinity because the fact is that the concept is something outside of human experience. God is the only multi-personal being there is, thus nothing else we experience in creation is exactly like him in this way.
The trinity doctrine teaches that the Son and the Holy Spirit are products of the Father.. they exist because the father exists. They "proceed" from the Father.
one of the difficulties in concieving of this is that humans usualy think of cause preceeding effect.. since God the Father effectively causes the Son and the Spirit to exist, we think of the Father as existing first.. but this is not the case.. they have all existed eternally.
The best example I have come up with for this is the image of a fire.. a fire emits heat and light. The heat and Light proceed from the fire.. it is their source, and their cause.. yet the fire doesn't really come before the heat and light.. the very moment fire exists.. heat and light also exist.
God has created everything, but only the Son and the Spirit actually originate from the essense of the Father himself. As a result they are of the same essense, or the same being.. They are God.
In modalism, God basicly puts on a play for humanity to reveal himself in three different ways or aspects.
In trinitarianism, God reveals himself throught he Son.. the Son is the aspect of God that we see. We do not see the father, nor can we experience him in our fallen state. The Son came to provide us a way to see the Father (through seeing the Son) and a way to be joined to the Father and have communion with the Father.
The Spirit acts as a link which joins us to the Son, and through the Son, we are able to come to the Father.