A citator is a tool that provides legal researchers with information:
The most popular legal citators are KeyCite by Westlaw and Shepard's provided by Lexis.
Prior History: What was going on with your case before it was decided.
Subsequent History: What has happened to your case since it was decided.
Citing Decisions: Cases that have cited your case.
Table of Authorities: List of cases cited by your case.
"In that respect experienced counsel such as Mazur's should know better than to cite Geise v. Phoenix Co. of Chicago--and if they did not know better to begin with, they should have learned better by the simple act of Shepardizing Geise (as every lawyer should do before citing any case)."
Horaitis v. Mazur, 2004 U.S. Dist. LEXIS 3066 (N.D. Ill. Feb. 24, 2004)
Citators are used for a number of reasons. One reason is to find a case or statute's direct history. With cases, one can find prior and subsequent history. With statutes, one can find whether the statute was amended or if there is pending legislation, for example. Another reason to use a citator is to identify negative treatment (such as if the case has been overruled) or to find positive treatment (such as another case that agrees with the analysis of your case). Citators can also be used to find additional primary and secondary sources. Finally, there are practical and ethical considerations. Lawyers must be thorough and prepared, and part of that involves making sure that the cases or statutes cited are still good law and can be used as precedent.
Failure to validate your research with citators means that you might use a case that is no longer good law in order to back up your argument. For students, this would result in losing points. For attorneys, this could mean losing the case for your client, or at the very least receiving a stern reprimand from a judge. For example, look at this clip of Judge Ito scolding Marsha Clark during the OJ Simpson trial for failing to properly Shepardize.
CALI.org has several CALI lessons to aid users in the use of Citators. Of particular interest are the CALI lessons on Updating/Validating Case Law Using Citators and Using Citators as Finding Tools.