It was first named Borderline because it was thought of as being on a line between psychosis and neurosis. The term is pretty out of date now and it has been considered by mental health bodies to rename it to Emotion Dysregulation Disorder, Unstable Personality Disorder, or Complex Post-traumatic Stress Disorder. The term neurosis is not widely used now. BPD is closely related to Post-Traumatic Stress Disorder. Basically the sufferer is so neurotic that the stress causes episodes of mild psychosis.
Neurosis
Someone who suffers from neurosis is basically extremely anxious very much all the time. They worry about all kinds of possibilities and have a general negative out look. The stress they feel is not socially accepted and is very challenging for the person and people around them.
Psychosis
When someone is suffering from psychosis they basically lose their sense of reality. They may suffer from a number of symptoms including audio or visual hallucinations, feel they have unrealistic abilities, and suffer from delusions. Some people with psychosis are a danger to themselves and occasionally others around them. It is not uncommon for someone with psychosis to be detained under the Mental Health Act.
Post-Traumatic Stress Disorder
PTSD develops after an emotional trauma. Many soldiers for example may suffer from PTSD. There are a number of triggers but the primary one is having your own or someone else's life put at risk. The disorder is consists of long term acute anxiety.