Alrighty, since it appears that I have not been thrown into a pit by angry literary scholars, it's time to continue with the summary of the Divine Comedy. In this installment, we're going to start getting into the actual circles of Hell and get to see how Dante has constructed his version of Hell. So, onward into the depths of the pit...

Inferno

Canto IV

Summary:

After recovering from his faint, Dante finds that he and Virgil are on the other side of the river Acheron. He and Virgil are at the edge of a huge funnel-like pit, which is how Dante envisions Hell. While traveling into the first circle of Hell, referred to as Limbo, Virgil talks about the inhabitants of said circle. The souls in the first circle are the Virtuous Pagans. Since these people died before salvation through Jesus was possible, the first circle is set aside for them, since they really couldn't help dying before the establishment of the church. He also mentions the Harrowing of Hell, which is when Jesus came down and pulled out the souls of important people from the Bible who died before being granted salvation. There really isn't a punishment in this circle except for knowing that they will never get to go to Heaven; otherwise, they are at peace. Eventually, they meet up with other great poets who are excited to see them and they enter the Citadel of Human Reason together. Dante is very excited to meet all of these great writers, but eventually he and Virgil must continue on and leave them behind.

Important Stuff:

Limbo = The edge of Hell, reserved for undamned souls that passed on before salvation was possible and for unbaptized children
Harrowing of Hell = This refers to when Jesus descended into Hell between his crucifixtion and his resurrection and freed the souls of the Old Testament saints. While Virgil is describing this, he refers to Jesus as the Mighty One because his name is not spoken in Hell.
Homer, Horace, Ovid and Lucan = All of these souls are considered by Dante to be the greatest of poets.
Dante mentions three types of souls in Limbo: Heroes (such as the Trojans, Hector and Aeneas), Philosophers (such as Socrates and Plato) and Naturalists (such as Ptolemy and Hippocrates)
There really is no punishment in Limbo in relation to the punishments of the other circles, but one way to look at it is the punishment of Limbo is knowing that they will never get to go to Heaven. So it's sort of a punishment, but nowhere near as severe as the others.
The garden in Limbo can be seen as that point in a story where the hero/traveller thinks it's okay to let their guard down when suddenly something bad happens, because Dante relaxes a bit while in Limbo with the other great Poets, but it only makes it worse when he encounters the first real punishments in the Second circle.
Being in Limbo is also a bit of a temptation for Dante, as he gets to meet what basically amount to his literary heroes, who made him "sixth in that high company." This is at odds with how he began this journey with a sense of humility.

Canto V

Summary:

The Poets have left Limbo and entered the Second circle, where they encounter Minos. Minos judges the sins of the dead and, using his tail as an indicator, determines which circle they must go to. Minos tries to turn Dante away, but Virgil declares that he can't because it is Dante's fate to see all of Hell. With that settled, they move on to see the punishment of the Lustful. The punishment for the Lustful is to be thrown about violently by a dirty, grimy windstorm for the rest of eternity. Dante asks Virgil to name some of the souls who are part of the storm and the Poet points out some notable figures such as Cleopatra, Helen of Troy, and Tristan, but Dante asks if he could talk to the shades of Francesca and Paolo. After calling the shades over, Francesca tells Dante her story. She was married, but fell in love with her brother-in-law Paolo. Eventually, they gave into their desires and had an affair, but her husband caught them and killed them. Dante, out of pity, asks what caused them to fall in love. Francesca tells him that she and Paolo were reading a poem about Lancelot and Guinevere and were so affected by it that they acted upon their desires. Dante is overcome by this woeful tale and passes out.

Important Stuff

The areas from the Second circle to the Sixth circle are deemed "Upper Hell", most of the sins punished in this portion are the sins of the she-wolf, also known as the sins of incontinence.
Minos = Based on the Greek king of the same name. Minos was the judge of the souls in the Underworld, so this is another example of Dante working classical mythology into his work. He is also know for his connection to the half man/half bull Minotaur, which is where I would guess he gets some of his physical attributes in this work, like the tail he uses to indicate which level a soul belongs. (The more times he wraps it around his body, the further down the soul goes).
"Hell's full lamentation" = The Second circle is where the actual punishment begins. Since Limbo isn't really there for punishing the souls, it's more of a place to keep them because they can't go to Heaven.
Reasoning behind the punishment = The lustful spent their lives being buffeted around by their chaotic lusts, so they are thrown around like that in Hell.
Francesca and Paolo = Francesca da Polenta got married to Giovanni Malatesta in order to stop the warring between the families, but Giovanni had some sort of deformity. Francesca fell in love with Paolo, her husband's younger (and healthier) brother, and they had an affair for ten years. Giovanni eventually caught them together and killed them.
Some people think that Francesca and Paolo being together in Hell is a romantic notion, but they are really there to remind each other of their shame.
Francesca mentions in the canto that "Caina waits for him who took our lives," showing that at the time of Dante's writing this work, her husband was still alive, but was destined for a deeper circle of Hell for killing members of his own family.

Canto VI

Summary:

Dante wakes up and sees that they have arrived at the Third circle. In the Third circle, a dirty slush storm rains down on the souls that are laying around in the putrid muck. These are the souls of the Gluttons. The three-headed dog Cerberus stands over them, drooling and snapping all over the place. Cerberus tries to attack the Poets, but Virgil is able to fend him off. A soul in the muck sits up as they pass by and says to Dante that he should know him because Dante was alive before he died. Dante, unable to recognize the ravaged shade, asks who he is. The shade is Ciacco, nicknamed the Hog by other Florentines, and he recounts how his sin was Gluttony. He also makes a political prophecy that details the events that will lead to Dante's exile from Florence. Dante asks about the fates of some other people who have died, and Ciacco informs him that they are located in deeper circles and, after asking Dante to talk about him with those who are still living, falls back into the sludge. Virgil tells Dante that Ciacco will not wake up again until Judgement Day. Dante asks if the final judgement of the souls in Hell will change their fates, and Virgil tells him that these souls will never be able to rise to Heaven, but they may get a little closer. They continue, picking their way through the swamp and head towards the edge of the Fourth circle, where they encounter Plutus.

Important Stuff

"I see new torments and new souls in pain
about me everywhere. Wherever I turn
away from grief I turn to grief again"
Dante still seems to be affected by the pain he sees being inflicted upon the souls, but since he believes that we must harden ourselves against sin and sinners, this one of the last times he will react this way.
Reasoning behind the punishment = Gluttons spend their lives wallowing in food and drink, and the only thing they created was waste; they are sentenced to wallow in waste and are left laying around like the garbage they created in life. Cerberus drools over them like they did over food while they were alive.
Cerberus = The three-headed dog that guarded the entrance to the Underworld in Greek myth. He symbolizes the Gluttons while they were alive, slobbering and tearing at the souls like they were something to eat.
Dante physics = "when our steps fell on a body, they sank through emptiness." Dante never really decides whether the souls in Hell are corporeal or not. In Limbo, Dante was able to walk over water like it was land and Virgil is able to touch Dante despite being a shade himself.
Political prophecy = Ciacco's prophecy is not actually a prophecy in the strict sense because it's basically a retelling of the events that lead to the writer Dante being exiled from Florence for his part in the politics there but to poem Dante, this would still be in his future, as this takes place in 1300 and writer Dante wasn't exiled until early 1302.
"Oh speak my name to the memory of men!" = Souls of the upper level want to be remembered in the realm of the living.
Judgement Day = After the resurrection of the dead during the Apocalypse, God will give a final judgement to all the people in the world. Ciacco won't wake up until this occurs.

And that's it: those are the first three circles of Hell. The next three cantos will see the end of the punishments for the sins of the she-wolf and the sins that are punished in Upper Hell.

Sources:

The Divine Comedy: The Inferno, The Purgatorio, and The Paradiso by Dante Alighieri, Translated by John Ciardi

Dante in Translation - Inferno I, II, III, IV (online Yale course available on Youtube): http://www.youtube.com/watch?v=QO2MPVQsHes

Dante in Translation - Inferno V, VI, VII (online Yale course available on Youtube): http://www.youtube.com/watch?v=5z2umV9UvIw

Harrowing of Hell: http://en.wikipedia.org/wiki/Harrowing_of_Hell

Minos: http://en.wikipedia.org/wiki/Minos

Francesca da Rimini http://en.wikipedia.org/wiki/Francesca_and_Paolo

The Last Judgement: http://en.wikipedia.org/wiki/Last_Judgment

Resurrection of the Dead: http://en.wikipedia.org/wiki/Resurrection_of_the_dead

Dante's Exile: http://en.wikipedia.org/wiki/Dante_Alighieri