* For anyone interested, I found a re-telling of the story in chronological order. It might help you piece together any sequences you found difficult to follow. Will poast below.
Here's what happens in real-world chronology. While things may seem confusing when you first watch the film, Nolan has been very careful to make sure that, when reassembled, everything in the main part of the film -- everyone's behavior and motivations -- makes perfect sense.
Leonard has been sitting around room 21 at the Discount Inn, poring over police files, trying to locate his wife's killer. He's talking on the phone, explaining his condition to someone on the phone. He relates the story of Sammy Jankis. Then he gets paranoid and hangs up the phone. But the person on the phone is persistent, even slipping notes under his door. The motel clerk finally tells him there's a guy, a cop, waiting in the lobby for him. Leonard relents and goes out to meet him. It's Teddy. We now understand that this is all a routine that Teddy has undergone with Leonard many times before.
Teddy's in the midst of a manipulative plan to have Leonard kill Jimmy Grantz, a local drug dealer. He gives Leonard the address of an abandoned building where Jimmy, who Teddy claims is the murderer Leonard is looking for, is due to arrive. Leonard, wearing blue jeans and driving a pickup, drives off, with Teddy following a few minutes behind.
At the building, Leonard kills Jimmy. He switches into Jimmy's clothes and takes his car keys. Teddy arrives and throws water on Leonard's triumph: You've already tracked down your wife's killers, he tells him; you just forgot. There's no such person as Sammy Jankis. Leonard's a mental case, Teddy tells him frankly. Teddy wants the $200,000 that he knows is in Jimmy's trunk.
The pissed-off Leonard decides to manipulate himself, setting up Teddy as his next suspect; he writes himself a note, identifying Teddy's license-plate number as belonging to his wife's killer. Leonard drives to the nearest tattoo parlor to get the number tattooed on his thigh. Teddy follows him there and tries to get Jimmy's car keys from him. (He wants that two hundred grand in the trunk.)
Leonard sneaks away, still wearing Jimmy's threads; by now he has no idea when or where he got these clothes or this spiffy car. But he finds a note in Jimmy's pocket and, assuming it's meant for him, he heads for Ferdy's bar to meet Jimmy's girlfriend, Natalie (Carrie-Anne Moss). Natalie sees the car pull up and is surprised that the driver isn't Jimmy. Leonard enters the bar. Natalie's heard of a guy with Leonard's condition hanging around. After testing his disability, in an unappetizing fashion, she's persuaded that he's is on the level, and takes him to her house.
After he watches TV and consults his notes for a few hours, Natalie returns. She surreptitiously hides all the pens and pencils in the room and then starts insulting Leonard, provoking him until he punches her. While Leonard desperately searches for some way to write a note to himself about what has just happened, Natalie goes outside, sits in her car and smirks. After a few minutes, she slams the car door, knocking Leonard's concentration off track, and reenters, crying about how someone named Dodd has beaten her up.
Moved, Leonard agrees to defend her from this supposed batterer. She writes a description of Dodd for him. He gets in the car to go after Dodd, but is immediately distracted: Teddy is waiting for him in the car. Teddy tells him not to trust Natalie and suggests that he stay elsewhere. He recommends the Discount Inn. Leonard has now forgotten about the Dodd business and, more amusingly, has also forgotten that he's already checked in at the Discount Inn, in room 21. Friendly, greedy desk clerk Burt gladly rents him room 304 as well.
Leonard sets up shop in 304 and calls an escort service for a hooker. He has her try to re-create the scene from the night he and his wife were attacked. He discharges her and drives to a trashy construction site, where he ruminates about his marriage and burns some of his wife's belongings. He stays there all night. As he leaves the construction site in the morning, Jimmy's car is spotted by Dodd -- a drug dealer who was Jimmy's boss. Wanting to know what's become of Jimmy -- and the money he was carrying -- Dodd gives chase.
Leonard slips away and goes to Dodd's motel room -- Natalie had given him the address -- and waits for Dodd to arrive. But he forgets where he is and why, assuming it's his own motel room. When Dodd shows up, Leonard mistakes him for an intruder and beats him up and tosses him in a closet. Desperate, he calls the only phone number he can find -- Teddy's. Teddy comes over and together they send Dodd packing. Teddy again makes efforts to get access to the keys to Jimmy's car.
Knowing from his notes that his run-in with Dodd had something to do with Natalie, the agitated Leonard goes back to her place, demanding an explanation. She placates him, agrees to help him identify the owner of the license-plate number on his thigh and takes him to bed. The next morning, they agree to meet for lunch, after Natalie has had a chance to look up the license number. Leonard forgets to take his motel key and leaves, but Teddy is waiting for him. They go have lunch, after which Leonard returns to the Discount Inn. Realizing he doesn't have a key, he asks Burt to let him in. Burt takes him to room 21 instead of room 304, and Leonard realizes he's being ripped off. But before Leonard returns to 304, he finds his note about having lunch with Natalie and dashes off to see what info she has for him. After some banter, Natalie gives him the DMV information, fingering Teddy as the killer -- just as Leonard had planned.
He goes back to his room and calls Teddy, telling him to come right over. At the front desk he tells Burt to let him know if Teddy shows up, but Teddy gets there while they're talking. Leonard drives Teddy out to the same location where he killed Jimmy -- having gotten the address from Natalie -- takes him inside the building and shoots him. It's the same shooting that we saw in reverse during the opening credits.