Memento is one of those movies that seems to make no sense. None of the characters are trustworthy. We cannot trust the main character since their memory is completely unreliable. Besides Leonard, Teddy would be the most knowledgeable character. His memory is in tact, but he often lies to Leonard to manipulate him.
So there is a dilemma, the narration of the film is unreliable and contradictory. It is also shown out of order to further complicate matters. Fortunately, there is an Easter egg hidden in the DVD release which will allow you to watch it in chronological order. While this doesn't explain everything, we can more easily see how everything fits together.
Paul's answer is partially correct, but has some details wrong. According to Teddy, Leonard often lies to himself to make him feel better. Teddy states that Sammy Jenkis was in fact a real person, but Leonard has the story mixed up.
Leonard did investigate Sammy's case, and believed he was a fraud. Sammy was in a car accident where he suffered a head injury. Leonard believed he was lying because Sammy would act like he recognized him when he came to the door. What Leonard did not realize at the time is that Sammy faked responses to act more normal.
According to Leonard's memories of Sammy's case, he had a wife that was a diabetic. Sammy's wife would routinely test his memory by hiding objects all over the house, and then she would try to get him to remember where he had left them. The strange thing about his injury is that he still was able to do complex tasks such as give his wife insulin. Since he had learned how to do that before the injury, he could still perform it. Now, his wife had become very depressed for some reason (we will get to that in a bit). One day, she tried in vain to get him to remember. She would call him repeatedly and have him do the insulin injection. Since Sammy could reliably perform this task, he did it every time she asked him to. Eventually, he gave her so much insulin that she fatally overdosed. Sammy was never able to fully reconcile what he did.
According to Teddy, Sammy actually had no wife. Sammy was real, but the story Leonard tells about him was not true. It was Leonard's wife that was diabetic, and it was Leonard that unwittingly killed her. Leonard's wife was also very depressed. Not only was she depressed about her husband, she was also depressed because of the attack. Whether or not she committed suicide is completely up for debate, but it was clear that she was not happy.
The psychology behind Leonard's character is very complex. He was diagnosed with a form of amnesia called anterograde amnesia. People with this form of amnesia have difficulty creating new memories.
Anterograde amnesia is a loss of the ability to create new memories after the event that caused the amnesia, leading to a partial or complete inability to recall the recent past, while long-term memories from before the event remain intact.
He also appears to have some other undiagnosed symptoms. Leonard also shows signs of having an illness called prosopagnosia, commonly known as face blindness. Prosopagnosia can also be caused by a brain injury, and it is likely he is also suffering from a form of it called associative prosopagnosia.
Associative prosopagnosia has typically been used to describe cases of acquired prosopagnosia ... People with this form of the disorder may be able to see whether photos of people's faces are the same or different and derive the age and sex from a face (suggesting they can make sense of some face information) but may not be able to subsequently identify the person or provide any information about them such as their name, occupation, or when they were last encountered.
This fact was alluded to due to the fact that he could look at a picture, and compare it to someone, but he cannot associate that image with any of his memories.
The only truth was at the end when Teddy confessed to Leonard, and he told him what really happened.