Default means to fail to perform some specific action that you are bound to do by law. For example, failing to make a debt payment. It is a very old word, originally meaning to sin or break the law. Here is what etymology online has to say:
early 13c., "offense, crime, sin," later (late 13c.) "failure, failure
to act," from O.Fr. defaute (12c.) "fault, defect, failure, culpability,
lack, privation," from V.L. *defallita "a deficiency or failure," pp. of
*defallere, from L. de- "away" (see de-) + fallere "to deceive, to cheat;
to put wrong, to lead astray, cause to be mistaken; to escape notice of,
be concealed from" (see fail). The financial sense is first recorded 1858;
the computing sense is from 1966.
It has nothing to do with bankruptcy, however, obviously if you are in financial trouble, you would default on your obligations, and that might well lead to lawsuits that would lead you to bankruptcy protection. However, they are not directly related in meaning.