Normal is compliance with the social etiquette and moral laws, and furthermore, the embracing of such things as one's centralized point for understanding oneself, and the society around them. It is a generally agreed upon set and series of actions and behaviours, to which the majority (typically the vast majority) follow.
To be abnormal is to express behaviours, opinions, or actions, which contrast to the generally agreed upon social norms of the society in question. I am relatively normal within the confines of where I live, but if you were to put me in China, or Iran, I would be committing social faux pas left and right.
To be deviant is to express behaviours, opinions, or actions, which the majority of people (or at least those who are in power over the society) consider to be illegitimate or immoral. An example of this in Western Civilization would be wanting to marry your sister. Some of these behaviours are accordingly punished determined by whether or not the society (or the people who are in power over society) decide if such behaviours cause damage of some discernible form.
Normal is a constantly evolving and changing state. Specific aspects of society are fluid, culture adapts and changes with technology, resources, politics, immigration, social trends, media, et cetera. Broad scopes of society do not change without extreme, sweeping mentality changes both on the part of the common person, and the government. (Ex: Women gaining the right to work and vote was one of the consequences of World War 1 & World War 2--when women were required to work in both wars to produce munitions for the front lines, thus ejecting tens of thousands of them from their social norms and forcing government to adapt to new norms itself.)
In terms of defining what is normal, that is not a particularly difficult task. You need only narrow the question to within reasonable guidelines, then within the frame of those guidelines, start asking questions.
For example: What is a normal set of behaviours, actions, and expectations for an adult male who grew up within Western Society in the modern era?
#1: They will get a job.
#2: They will pay taxes.
#3: They will be more likely than women to be the main breadwinner, and to feel compelled to be the main breadwinner should it come down to a choice between them.
#4: They are more competitive than their female counterparts.
#5: They are more likely to die in workplace accidents or wartime fatalities.
#6: They will generally abide by empathetic laws, so long as they are not greatly intoxicated or mentally unstable. (Ex: They will not murder, they will not commit grand theft auto, et cetera.)
Those are six off the top of my head. You can then further extrapolate and attempt to explore the root cause for why these are normal behaviours and actions and expectations, in contrast to what would then be considered abnormal behaviours, actions, and expectations. For example...
"Why do men get jobs?"
A. To be self-sufficient, which is an aspect of masculinity--culture.
B. To be able to acquire goods and services which they desire--ego, survivalist.
C. Because society looks down upon men who do not attempt to earn their keep--social pressure.
Et cetera.
tl;dr: Occam's Razor and the Scientific Method are your friends in the quest to understanding anything in the universe.