Almost all of us spend some portion of our lives wondering if we're normal. Not a bad question, since the National Institute of Mental Health (NIMH) estimates that more than 1 in 4 Americans have a mental disorder. Someone with a mental disorder has behaviors, feelings and thoughts that deviate from the norm. In these circumstances, someone who isn't "normal" doesn't match up to what society (and medicine) considers standard. When we ponder what's normal, it's often in the sense of determining whether the way we think and act is the same as -- or at least similar to -- the majority of other people.
In a social situation like a party, for example, you probably make polite conversation with a stranger about the host, the food and weekend plans. But a person with Asperger's syndrome might have a 30-minute conversation with someone about engineering, not noticing bored expressions or restless shifting from foot to foot. This behavior is outside established parameters of what's considered acceptable, or normal, social behavior.
When it comes to evaluating our own behavior, we usually decide how to act based on our own perception of what's normal. If you gamble compulsively and feel it's an unusual behavior -- that you're one of a very few people in the world who can't stop gambling -- you might feel ashamed and hide it. But if you learn that millions of people in the United States struggle with gambling addiction and support and treatment are available, you might be more likely to try to change behavior.
At the same time, social standards have a strong influence on idea of "normal." A person Western society views as a gambling addict might be seen as a typical adult in a culture that considered unstoppable gambling to be a reasonable behavior.
We can look at a society's laws to see what's culturally considered normal, but that doesn't always give us the right clues. For example, smoking marijuana recreationally is illegal in the United States, yet millions of Americans have indulged.
In another sense of the word, normal means average or standard. So while an alcoholic may yearn to lead a "normal" life, a bored high school student may yearn to lead anything but. Normal, seen through the eye of the beholder, is filtered through the lens of society.