SawIV

Jigsaw's autopsy opens the film. Very graphic. The wax tape from the last movie is found in Jigsaw's stomach. Detective Hoffman is notified, arrives in the medical examiner's room and he plays the tape, Jigsaw promises even after death, his games will continue as his message.
First trap is shown, two men are connected to locks and a chain to a pulley device in the middle of a room somewhere. One has his eyes sewn shut, one has his mouth sewn shut. They are unable to communicate with each other. The keys to their freedom is on the back of the other's neck. The device is started and it slowly pulls them both closer as the chain is wrapped around an axle and it forces them to fight to the death (with various weapons to use), otherwise the machine will kill them both. The mute guy kills the blind guy by bludgening him to death with the back of an ax. Afterwards, he uses the key that he took earlier during the melee and opens his lock. He rips his stiches in the process and his lips bleed profusely.
(There was no TV or tape message for the two men to inform them why they were there or how to escape their trap.)
SAW IV title appears
Detective Kerry's body is found where she was killed by the police task force led by Detective Hoffman. Sgt. Riggs is part of the team and he breaks protocol by rushing to Kerry thinking she was still alive when her face is shown to the unit by a bomb rover robot that was sent in first. He is scolded by Hoffman for doing that. They are greeted by FBI agents Perez and Strahm. Kerry was their liason to the police department. The agents go over the crime scene and deduce that Jigsaw was not physically able to to get Kerry up on the Angel of Death trap because of his bed ridden condition and neither was Amanda because she was not strong enough. There's a third person involved. Hoffman sees that Riggs is emotionally wrecked by discovering his dead friend and sends him home.
Riggs arrives home to see his wife leaving him because he's never home due to his duties as a police officer. He takes the break-up like a champ and goes to bed after she leaves. He wakes up after hearing the door open. Thinking its his wife, he gets up and looks around his blackoutted home before being attacked by The Pig Man. He wakes up in his own shower that's on and realizes he's part of Jigsaw's game when he sees the cryptic red messages smeared on the wall. He goes into the next room and activates the TV informing he has 90 minutes to save the lives of Detective Matthews and Detective Hoffman (who was kidnapped shortly after he sent Riggs home), but along the way he will tested on whether or not to save the lives of other victims of Jigsaw's game (ala Saw III). In the next room, he finds a woman chained a chair.In the tape recorder next to her, Jigsaw informs that Riggs can make the choice to save her or walk away before the time expires.
Second trap: The timer starts and the woman's hair is bound to a turning axle that is slowly ripping her scalp off. Riggs tried to cut her off, but all his sharp and cutting tools have been removed from his house. He tries shooting the gears, but that doesn't stop the machine. He saves the woman by unlocking the combination to free her after discovering the numbers on the gears off the machine. While he isn't looking, the woman named Brenda grabs a knife that was hidden and tries to kill him. Shocked and confused, Rigg is forced to kill her. She had a tape player on her that informed her that if she didn't kill Rigg, she would be going to prison for life because there was incriminating evidence of her placed in Rigg's home.
Rigg's next test leads him a motel where the front desk clerk is the next subject. Jigsaw left Riggs two keys. One is for a motel room that is filled with pictures hung on the wall of battered women and a bed that has shackles attached to it.
The clerk (forgot his name) comes into the room because he wanted to confront Rigg about not checking in. The clerk turns out to be a rapist in Jigsaw's next tape player who enjoyed taking pictures of all his female victims as well as videotaping his forced sexual acts with them. Rigg forces him at gun point to harness himself in the bed shackles.
Third trap: The clerk will be killed if he does not gouge out both his eyes within the time limit. He manages to take out one after pushing a button for a mini-scythe to plunge into his eyeball, but he cannot take out the other and he is killed in brutal fashion (his death was pretty much censored for the most part, but it looked like all his limbs were lopped off by bigger scythes latched over the bed). Riggs leaves to his next test location.
Rigg is led to a classroom where a man and a woman are tied up with each other's backs touching. They were both impaled by sharp metal poles, but the woman is still alive. Her test was to pull out the poles, otherwise she would bleed to death. The man was her abusive husband and he was a goner anyway because the poles had penetrated vital arteries in his system. He was only still living because the poles were stopping massive bleeding, but the wife decided to save herself and he dies because of this. Rigg has the key to her chain and gives it to her because she is only able to free herself as Jigsaw states in the tape message there. Rigg leaves to go take the final test.
All while these events are going on, Agent Perez and Strahm comes across the trap crime scenes one by one. They debate whether Rigg is a victim of Jigsaw's newest game or an accomplice (has incriminating evidence piling against him with his disappearance and dead bodies showing up). They are informed that a doctor (Lynn) from a hospital and Detective Hoffman are missing.
Eric Matthews had been missing for 6 months. (Kerry's first name is revealed to be Allison by the way, forgot to mention that earlier.) Following the sewer fight with Amanda, Matthews was abducted by a hooded man, put in a cell alone, fed, and his broken foot held in placed by a metal brace that was attached by the man during this time frame. In present time, his trap has him in a noose over the block of ice that will melt once the 90 minute limit is up.
Detective Hoffman is bound in a chair next to Matthews. Their traps are on a large tipping scale, which means they are connected to each other. If Matthews' body is off the block of ice that is holding him up, not only will he be hung, but Hoffman will be lowered into water and electrocuted (wires are placed from a power grid) nearby if the weight difference reaches a certain level.
A man is in the same room as them and he is watching all the tests of Rigg through several monitors and it turns out to be the guy that survived from the trap from the beginning of the movie. His name is Art. He was Jigsaw and Jill's lawyer, but he also represented the rapist clerk (got him cleared off three rape cases) and the abusive husband.
The abusive husband also gave his young daughter the same treatment as his wife and lied to the police about it when he was questioned at the station. This was when Rigg was still new to the force. He attacks the man and is facing a lawsuit for the incident, but Hoffman sticks up for him and lies to Art saying the guy attacked Rigg first.
We are led to believe that Art is Jigsaw's replacement, but not so as it is revealed that he is just the new Zepp (in a game of his own and forced to help carry the game out). He has a weird contraption strapped to his back that will probably kill him if tampered with or if it reaches a time limit. it is not explained how he got that on him, how the contraption works, or why he is in another game when he survived the last one. He gives Matthews a gun with one bullet and Eric has to shoot whoever comes through the front door before they open it otherwise he will be killed by giant ice blocks coming down and crushing his head (think the log trap from Return of The Jedi).
Art also has an electronic box with a red button on it that when pushed him, him and Matthews will be released from their trap, but Art cannot push the button until Rigg's timer expires.
Agent Perez and Strahm track down John's ex-wife Jill and bring her to the station to be interrogated. It turns out John was an engineer who was head of his own company and ready to settle down with Jill and be a family man, but left that all behind him following the death of his first and unborn child along with being diagnosed with cancer when he attempted suicide.
Having survived trying to take his own life, John became Jigsaw and tracked down the man responsible for murdering his child. (The man is Cecil and he was a drug addict and thief. He causes the baby's death when he slams a door into Jill's pregnant stomach when he returns to Jill's clinic, where she worked, after closing hours to retrieve his coat. John witnessed him running from the scene as he was waiting to pick up Jill from work.)
We see Jigsaw taking inital steps to his iconic role as he is trying out different test messages with his altered voice as well as designing and building the traps. He cleverly abducts Cecil in public in broad daylight during a Chinese festival and brings him back to his workshop in development. Cecil is Jigsaw's first test subject.
Original trap: Cecil is bound to a chair. His wrists and ankles are all stabbed through with knives (Cecil had a thing for knives) and to be released from the trap before he bleeds to death, he must push his face as hard as he can through even more knives. He does this after several failed attempts of not applying enough facial pressure, is cut up pretty bad, but he does not learn his lesson and tries to kill Jigsaw, but misses when Jigsaw sidesteps and Cecil falls into a different trap meant for someone else and gets wrapped up in barb wire and dies.
 
Jill notices John is becoming more distant and volatile since the baby's death and suicide attempt. Jigsaw breaks up with her.
Agent Strahm gets furious with Jill during interrogation because she has incriminating evidence against her mainly due to her past relationship with John, but she maintains her innocence.
Agent Perez gets injured by a booby trap (grenade hidden inside Billy The Puppet's head) during the class room crime scene search. The wife managed to free herself and survived.)
Agent Strahm figures out Rigg is going to the warehouse (where the final test is) that Jigsaw owned and tries to intercept him there. Agent Strahm arrives there and enters the warehouse through a different entrance than Rigg.
Final test: Rigg enters the warehouse and opens the door to the room Matthews, Hoffman, and Art are in. Not knowing who is coming through, Detective Matthews takes aim and fires, only wounding Rigg in the shoulder. The trap is sprung and Matthews dies a sh**ty death as his head is smashed by the two big ice blocks coming down from opposite directions, not even knowing what happened to his son Daniel or Kerry. Rigg uses the last of his bullets to kill Art, thinking he was behind it. A tape recorder is there for Rigg to play. "Hello Zepp" plays and Jigsaw informs him that he failed his game since the point was to save Eric and he caused Eric's death instead. The point of the game was that according to Jigsaw's belief. People cannot be saved by others, but only by themselves. And Rigg's obsession with wanting to save people backfired on him. Had Rigg waited a second longer, Art could've pushed the button and they would all be alive.
Hoffman unties himself and walks over, it is revealed that he was the other apprentice of Jigsaw (like Amanda, he was pretending to be a game victim). He wasn't kidnapped. We just assumed he was. He was the one who kidnapped Kerry and Dr. Denlon as The Pig Man. He helped put Kerry in her death trap since he was strong enough to lift her, he wrote the letter in Amanda's envelope, but the contents of it were still NOT revealed! !@#$%^&*, he also knocked out Rigg at his house and helped set up his game. Hoffman leaves Rigg in the room.
It is revealed the events of SAW IV are taking place immediately following SAW III by a few moments. The warehouse Rigg and Strahm are in is the location of SAW III. Strahm winds up in Jigsaw's lair hearing the ending conversation of SAW III with Jigsaw talking to Jeff then the buzz saw, explosion, and the metal door closing. He opens it and sees Jeff holding a gun. Jeff thinks Strahm is involved and asks where his daughter is before being shot to death.
Strahm looks around and sees the bodies of Jigsaw, Amanda, and Lynn. Hoffman comes from where he was and closes the metal door locking Strahm inside. Hoffman turns the lights inside the room off and walks away.
We cut back to the Autopsy Room with Hoffman listening to Jigsaw's tape from the beginning. We assumed Jigsaw left it for any detective, but it was specifically meant for Hoffman to listen to.
The tape says...
"Are you there detective? If you are listening to this, then you must be the last man standing You feel you now have control, don't you? You think you will walk away untested. I promise that my work will continue. THAT I have ensured. By hearing this tape some will assume that this is over but I am still among you. You think it's over just because I am dead. It's not over, the games have just begun."
Jigsaw's face is the final shot before cutting to credits.