The typical first-year student walks into his first economics class with very little idea of what economics is. He might have heard something like, "economics is the study of money", or "economics is another word for accounting", or "economics is hard, don't take that class", but none of those are true.
"Economics is the study of the use of scarce resources that have alternative uses." That's the classic definition of economics. Basically, there are people, and people need resources to fulfil their desires. These resources cannot be infinite, but the desires can be, so people need to make choices about how to use their scarce resources. Economists study these choices.
All economic questions fall into one of two categories: positive and normative. Positive economics describes "what is" and normative economics argues for what ought to be, so a question like, "why do people use money?" is a positive question and "should people use money?" is a normative question.
A general rule of thumb is that if your economic model has no value judgements, it's positive economics, and if it does have value judgements it's normative economics, since to tell someone what he ought to do, you first have to judge what is best for him.
Economics is also divided into microeconomics and macroeconomics. Microeconomics studies the behaviour of individual agents and markets, while macroeconomics studies the behaviour of the entire economy.
Economists also have their own branch of statistics called "econometrics" that's specialized to analyzing economic data. Since economic data usually comes from the real world, and not from controlled experiments, econometrics faces mathematical challenges that other fields might not.
The tools economists have developed to study human behaviour have broad uses outside of what we would traditionally consider economics. Economists study not only markets, but things like crime, war, the family, religion, culture, politics, law, and even genetics. That's why it's not unusual to see papers by psychologists, sociologists, criminologists, political scientists, anthropologists, biologists, neuroscientists, or legal scholars being co-authored by economists.