I’ll focus on Christian concepts first, then move on to Jewish concepts (which are shared to some degree by Christians, though Jews reject the Christian ones)
Heaven is “where” God is in his fullness and being there is the state of union with Him. This is the desired destination for all Christians.
Paradise is the same thing, “Paradise” just being the Latin-derived term while “Heaven” comes from Anglo-Saxon. Indeed in many Romance languages the word for heaven is “Paradiso” or something similar.
Purgatory is in Western Catholic tradition the intermediate state for those souls destined for Heaven but who are not yet completed purged of sin prior to entering Heaven itself. In the middle ages people sometimes thought of it as basically a temporary version of Hell, in that you would receive punishment for your sins, though only until you had suffered the due penalty for the sins you had died with. More recently, it’s gradually moved in many circles to an understanding more similar to the Eastern Orthodox concept of Hades.
Hades, then is the Eastern theology concept of the intermediate state. Basically, it’s sort of the waiting room for the Day of Judgment. For the righteous, it’s a joyous time, as you know that soon you will be in Heaven forever with God. For the wicked however, it’s torment, as you are confronted with your sins and have nowhere to hide from the sight of God (think Adam and Eve hiding from God after eating from the tree)
Limbo is a Western medieval concept based on the proposition that before Christ’s resurrection Heaven was closed to everyone, righteous or not. But since it was obviously improper to imagine the saints of the Old Testament being punished in Hell, the concept of Limbo was created where the righteous of the Old Testament were forced to wait for the resurrection before entering Heaven, but were spared the sufferings of Hell. A popular medieval image was the “Harrowing of Hades” (no relation to the Eastern concept of Hades above) in which Christ after his death and descent to Hell freed all the souls of the righteous kept there, who finally entered heaven when Christ was resurrected. Recently theologians, including Pope Benedict XVI have downplayed the concept of Limbo.
Finally, Hell is the destination for the damned-- the souls of those who reject God and his offer of salvation. This is the “lake of fire” Jesus spoke of, where there is “wailing and gnashing of teeth” for all eternity. It is best understood as total separation from God.
Finally the Jewish ideas:
Sheol was the abode of the dead. Jewish theology especially before the time of Christ was full of speculation as to what came after this life. Some (such as the Sadducees) didn’t even believe there was a resurrection or continuation of any sort. But more generally it was conceived as similar to the idea of Hades above.
The Bosom of Abraham, in the same theme, was a special condition in Sheol for the righteous. God would reward their lives of devotion and faithfulness by placing them in a pleasant and peaceful state. This is somewhat comparable to the Christian idea of Limbo seen above.
Hope this explains things!