Well first off, not every bit of the story is explained in the albums, for example, there's no song depicting the fall of House Atlantic in YOTBR or Mayo telling Coheed about the Monstar/Sinistar in SSTB.
SSTB:
Time Consumer-Coheed and Cambria killing their children.
Devil in Jersey City-CoCa's daughter, Josephine, and her boyfriend, Patrick, are out driving. Patrick is about to propose, but they are attacked by a gang called the Jersey City Devils.
Everything Evil-Josephine and Patrick run back home. Coheed kills Josephine with a hammer, Patrick runs away in fear. Claudio comes home after Mayo and the Red army capture CoCa, his Crowing powers briefly start to awaken, and Josephine gives Claudio some last words.
Delirium Trigger-CoCa start to fight back aboard the Red Army's ship.
Hearshot Kid Disaster-Claudio has snuck aboard a ship to get off-world, lands on a planet where the red army is throwing an alien race into the keywork. One of those aliens tried to rise up, and is known as the "Hearshot Kid."
33-Patrick has been tracked down as a witness to the murder of Josephine and is killed himself by one of Wilhelm Ryan's Ostantine Priests, mages who have been turned into horrible monster mind-slaves.
Junesong Provision-"Junesong" is Newo Ikkun (Claudio's girlfriend's) nickname. This is the note Claudio leaves her before he leaves.
Neverender-I'll have to get back to you on this.
God Send Conspirator-Maria, Jesse's wife, is killed trying to stage her rebellion against Mayo.
In Keeping Secrets
Ring in Return-Signifies the time Claudio has been sleeping on the planet Shylos 10.
IKS-Jesse is reforming Maria's rebellion. One of his IRO-bots, Sizer, is jealous of Chase, a IRO-bot who can hear God's thoughts.
CMitMoM-Chase is kidnapped by the Red Army, but Sizer gives himself up instead to save her.
Three Evils-Sizer is tortured for information, but he doesn't give in. The Red Army disassembles him, where his remains are found by Claudio. They are then found by Jesse.
The Crowing-Ambellina finds Jesse and Claudio and tells Claudio he is the Crowing, the one God has sent to destroy what he has made.
Blood Red Summer-Jesse recounts to Claudio what has happened to his parents and what he must do now.
Faint of Hearts-The scooby gang needs to get to House Atlantic to destroy Willhelm Ryan. To slip in, they enlist the services of Al the Killer, a mercenary with a fetish for sniper rifles.
Backend of Forever-Al falls for Ambellina, and see's her avoidance of him as rejection, so he swears never to love again.
Al the Killer-Frustrated, Al decides to betray the group to Ryan.
A Favor House Atlantic-As Al hands the gang over, he feels that he is the only one who should kill Ambellina, and helps the group escape in a heroic last stand.
The Light & the Glass-Claudio reminisces about the time he shared with Newo and how he wishes things could be different.
21:13-Claudio has a nightmare about the events of the TSSTB.
Good Apollo Vol. 1
Always and Never/Welcome Home-The Writer learns that his girlfriend, Erica Court, has been cheating on him. He dreams about killing her in love and anger.
Ten Speed-The Writer begins to imagine his rage in the form of his ten speed bicycle, who tells him to kill Ambellina to get over his grief and to help his character (Claudio) awaken his true crowing powers.
Crossing the Frame/Apollo I/Once Upon Your Dead Body-As far as I can tell these are just lots of different ways the writer imagines getting revenge.
Wake Up-Despite all his rage, the Writer still loves Erica.
The Suffering-The Writer imagines proposing to Erica.
Apollo II-In this song, Jesse is killed by Mayo, and the Writer comes through into the story world through the Willing Well as the group, along with Jesse's army, is attacking House Atlantic. The Writer kills Ambellina in order for Claudio to awaken his powers as The Crowing.
The Final Cut
Good Apollo Vol. 2
NWFT-Claudio and the other IRO-bots begin the final battle at House Atlantic. Ryan vows to kill the Crowing.
The Hound-I'm not entirely sure about this one, or who the hound is (Claudio stated that he is a new character).
The Running Free-Sizer is killed by the Red Army, but finds peace in his death.
Feathers-The negative feelings of Erica from the last album begin to creep into Claudio's thoughts in the story towards Newo. In this song he officially cuts off all ties he had to the world so that he can carry out his duty.
Mother Superior-Claudio is consulting the memory of his mother to help him carry out his goal of destroying Heaven's Fence.
Gravemakers and Gunslingers/Justice in Murder-Claudio and Mayo face off in a gunfight, and Claudio has finally avenged his parent's death. He also is able to rationalize the future murder of Wilhelm Ryan.
Radio Bye Bye-Claudio goes in to face Ryan alone.
The End Complete-Claudio and Ryan fight each other.
The Road and the Damned-The Prise try to convince Claudio not to destroy the Fence, but he does it anyway.
On the Brink-Heaven's Fence falls, and the story ends.
I'm going to assume you've read the YOTBR book and I don't need to explain it to you. Clearly I don't have anything better to do than explain (almost) everything song by song.