Economics is the study of how people and societies allocate resources. Everything, from money to clean water to oil to iPods and Barbie dolls, is finite. It is impossible for everyone to have everything. Economics is the study of how people get the things they want and need and how these things are distributed. When teaching economics, the objective is for students to understand these concepts from a wide variety of angles.
The objective of microeconomic education is to teach students about how people make choices at the individual or household level. Microeconomics also teaches students how these people interact in markets: how your decision to pay higher-than-average rent will ultimately affect your neighbor, for example, by marginally bringing his rent up, too.
When someone finishes a microeconomics course, he should have a better understanding of how his economic actions effect the people around him.
The objectives of teaching macroeconomics are the same as microeconomics, just on a larger scale. Macroeconomics studies the effects of decisions made by groups of people, rather than the effects of decisions made by just a few people. Inflation, unemployment, and exchange rates are all parts of macroeconomic theory -- the combined effect of aggregate decisions.
The objective of teaching macroeconomics is for students to ultimately understand how the large forces that affect them come about. Someone who understands macroeconomics will understand, for example, how a strong currency (negatively) affects exports. What's more, he'll understand what made that currency strong in the first place.
Econometric studies are the application of the two theoretical branches mentioned above. Rather than just understanding vaguely how things should work, econometric study is about looking at real data and analyzing it to draw conclusions.
A student who finishes a course of econometric study should understand how to examine data, how to draw conclusions from it and what it means. What's more, he should be able to back up these conclusions with firm arguments from micro- and macroeconomic perspectives. Econometrics, then, is best taught to students who are already familiar with the other two perspectives.