David, Post Traumatic Stress Disorder (never says this in the film but his flashbacks, nightmares, anxiety, rumination, hypervigilance etc all point to PTSD... plus this was the 50s and PTSD didn't have a name then).
Donald Morton and Isabelle Sorenson, Asperger's Syndrome. Asperger's is a PDD (Pervasive Developmental Disorder) and can be thought as a high-functioning form of autism (for the sake of keeping this note simple).
Conrad Jarrett, Major Depressive Disorder and Acute Stress Disorder (possible PTSD depending on how long ago the death of his brother was, the difference between ASD and PTSD is the duration of symptoms since the trauma).
Sarah Connor, diagnosed with schizoaffective disorder (traits of bipolar and schizophrenia) by the idiots at the Pescadero mental institution. Remember, just because someone is diagnosed with something, doesn't mean they have it.
Given John Connor's history of theft, vandalism, truancy, etc it's not unreasonable to note that a lot of shrinks might diagnosis him with Oppositional Defiance Disorder, if not outright Conduct Disorder. I can even see some over-zealous shrinks trying to diagnosis John with Reactive Attachment Disorder because of Sarah's "behavior" while he was a baby/toddler. Yes, sometimes a cigar is just a cigar.
Campbell. Never explicitly stated but he behaves like someone with Post Traumatic Stress Disorder.
A list of movies dealing with mental illnesses or psychological issues. When the illness isn't stated in the movie I have guessed to the best of my abilities (I have studied psychology but I am not a psychologist). This list is for entertainment lists only, no offense is meant to anyone. However, if you are interested in a certain disorder, you may want to check out the movies where a character suffers from said disorder in the list.