A citator is a "catalogued list of cases, statutes, and other legal sources showing the subsequent history and current precedential value of those sources. Citators allow researchers to verify the authority of a precedent and to find additional sources relating to a given subject." - Black’s Law Dictionary, 8th ed.
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. Another reason 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 cited are still good law and can be used as precedent.
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.
Failure to validate your research with citators means that you might use a case that is no longer good law in order to support your argument. For students, this would result in losing points on an assignment. 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.