Shepardizing refers to checking other cases that cite a case (or a piece of legislation), to make sure it's still "good law". You don't want to cite a case that's been later reversed or overruled, or a law that's been found unconstitutional. It's called Shepardizing after Shepard's Citations, a publication that would list the authorities that cited a particular case.
"Shepardizing" is a trademarked term used in LexisNexis. In Westlaw they call it KeyCiting. It includes symbols that indicate how a case has been treated. At the top of the case there may be a red or yellow flag or symbol. Red (stop sign) means that the case, or part of it, has been reversed or overruled. Yellow (yield sign) means it's been treated negatively.
Westlaw's Guide to KeyCite (PDF)
