Experience has shown that as software is developed, this kind of reemergence of faults is unfortunately quite common. Sometimes it occurs because a fix gets lost through poor revision control practices (or simple human error in revision control), but just as often a fix for a problem will be "fragile" - if some other change is made to the program, the fix no longer works. Finally, it is not unknown that when some feature is redesigned, the same mistakes will be made in the redesign that were made in the original implementation of the feature.
Therefore, in most software development situations it is considered good practice that when a bug is located and fixed, a test that exposes the bug is recorded and regularly reperformed after subsequent changes to the program. Some projects (Mozilla, I think does it) even set up automated systems to automatically re-run all regression tests at specified intervals (usually daily or weekly) and report any regressions. Or, if you are using make, you could set up your makefile to automatically run the regression tests after every successful compile.
Regression testing is an integral part of the extreme programming software development methodology. There design documents are replaced by extensive, repeatable and automated testing of the entire software package at every stage in the software development cycle.
Remaining questions - who invented the term, where did it come from - academia, industry, ???? Who was the first to use it systematically? Does it get a mention in the Mythical Man-Month, for instance?