In pure mathematics, "well-behaved" objects are those that can be proved or analyzed by elegant means to have elegant properties.
In both pure and applied mathematics (optimization, numerical integration, or mathematical physics, for example), well-behaved also means not violating any assumptions needed to successfully apply whatever analysis is being discussed.
The opposite case is usually labeled pathological. It is not unusual to have situations in which most cases (in terms of cardinality) are pathological, but the pathological cases will not arise in practice unless constructed deliberately. (Of course, in these matters of taste one person's "well-behaved" vs. "pathological" dichotomy is usually some other person's division into "trivial" vs. "interesting".)