Psychology is the study of the human brain and human behavior, so "Normal Psychology" - usually referred to simply as psychology is the study under typical (typical is how we determine what is "normal" or normed behavior) human conditions. So the study of conditioned responses, positive and negative reinforcement, physical maturation, etc. could all be classified as normal psychology.
"Abnormal Psychology", then, is the study of the human brain and behavior under atypical conditions. Someone with schizophrenia, for example, does not think or behave normally - the study of that person's behavior is abnormal psychology. The behavior of sociopaths, criminals, and the more mildly deviant also fall under this category, as does abnormal or deviant sexual behavior.
It is very difficult to differentiate between normality and abnormality. That's one of the major problems psychologists have in diagnosis and that's also one of the reasons there are so many editions of diagnostic manuals, like the DSM. What's 'normal' in one culture or situation, may be completely 'abnormal' in another.
There are 7 basic criteria for defining abnormality. Disorders don't have to match every single one of them, but should match two or three.
1. Statistical infrequency (if it's uncommon - however, depression is quite common)
2. Maladaptiveness (the disorder doesn't allow you to function in terms of 'play, work and relationships')
3. Social norms (doesn't fit into social norms)
4. Irrationality (there's no rational explanation)
5. Suffering (psychological suffering - however, if no other category is matched as well, then the condition is not abnormal)
6. Unpredictability and loss of control (ordinary pattern of behaviour is broken + cause for action can't be identified)
7. Observer discomfort (those watching are uncomfortable)
Abnormal Psychology then deals with diagnosis and therapy of those who are abnormal, while normal psychology is understanding the cognition and behaviour of those who are not. Different approaches (psychodynamic, cognitive, behavioral, biological, humanistic etc) can be taken for both types of psychology.
In normal psychology a person appears to be in good mental and physical health. He tends to be conforming to societal expectations and would be generally considered to be a normal person by the general population. The patterns of behavior tend to be parallel to those demonstrated by most of society.
Abnormal psychology is more relevant to the things that are weird or different. Most people are aware that hearing voices telling them to kill someone or nail themselves to a cross would be abnormal unless one is abnormal. Abnormal psychology is when the function of the person's mental abilities becomes so abnormal that a person can not function as a member of his society in an expected social manner. The patterns of a person's behavior tend to be different than those of the people who are in a particular society.
Normal psychology is the study of the human brain and human behavior under normal conditions, where the human does not have a disease or disorder. Abnormal psychology is the study of the human brain and human behavior under abnormal conditions, where the human has a maladaptive, harmful dysfuntion as a disease and/or disorder.