The ones you need to know about are four:

1. Half Kaddish (Chatzi Kaddish)

This is just the first and main part of the Kaddish. It is said only by the leader of the prayers. The only exception is after the Torah Reading, when a mourner may be asked to recite it.

The great Kabbalist, Rabbi Isaac Luria ("The Ari"), explained that the various sections of the prayer are like the rungs of a ladder climbing upward—just like the ladder Jacob saw in his dream. The Kaddish, he said, is placed strategically just before the ascent to the next rung to give us the power to move up. Once we get all the way up (in the silent prayer called the "Amida"), we need the Kaddish again as we descend back down—in order to stay connected to those higher realms even as we go about life in our mundane world.

2. Full Kaddish (Kaddish Shalem)

This includes the following lines:

May the prayers and petitions of all the House of Israel
be accepted before their Father in heaven,
and they should say Amen.

May abundant peace and good life
come down to us and to all of Israel from heaven,
and they should say Amen.

May He who makes peace above
make peace upon us and upon all Israel,
and they should say Amen.

This is said after we've finished asking G_d for our needs in the silent, standing prayer called the Amidah. It is always said by the leader of the prayers alone.

3. Mourner's Kaddish (Kaddish Yatom)

This is the same as Full Kaddish, only that the paragraph asking for acceptance of our prayers is skipped. It is said after saying Psalms or any other hymns to G_d with a minyan. All the mourners in the synagogue recite it together. If there is no mourner, the leader of the prayers says it. However, often the leader of the prayers will refrain from saying it, since it is customarily only recited by someone who does not have both parents alive.

4. Rabbi's Kaddish (Kaddish D'rabbanan)

This is the same as Full Kaddish, only that instead of the paragraph asking for acceptance of our prayers, we say the following:

Upon Israel and upon their rabbis and upon their students
and upon all the students of their students
and upon anyone who is occupied in Torah
they should have and we should have
abundant peace, grace, kindness and compassion
and long life and plentiful sustenance
and liberty before their Father in Heaven
and they should say Amen.

This is to be said anytime ten Jews learn Torah together. Since there are several places in our prayers that are actually not prayers but teachings, this Kaddish is said then. All the mourners in the synagogue recite it together. If there is no mourner, the leader of the prayers must say it.