Meters can be classified by counting the number of beats from one strong beat to the next. For example, if the meter of the music feels like "strong-weak-strong-weak", it is in duple meter. "strongweak-weak-strong-weak-weak" is triple meter, and "strong-weak-weak-weak" is quadruple. (Most people don't bother classifying the more unusual meters, such as those with five beats in a measure.)
Meters can also be classified as either simple or compound. In a simple meter, each beat is basically divided into halves. In compound meters, each beat is divided into thirds.
A borrowed division occurs whenever the basic meter of a piece is interrupted by some beats that sound like they are "borrowed" from a different meter. One of the most common examples of this is the use of triplets to add some compound meter to a piece that is mostly in a simple meter. (See Dots, Ties, and Borrowed Divisions to see what borrowed divisions look like in common notation.)