First, and most commonly, a semi-colon connects two independent clauses.
(As a quick refresher, an independent clause is a phrase that can stand on its own as a sentence. For example, “The French bulldog puppies scrambled for the new toy” is an independent clause, but “Whenever the French bulldog puppies scrambled for the new toy” is not.) Though these clauses may lack a coordinating conjunction between them—“and,” “but,” “or,” “nor”—they should be related in meaning. For example, “The woman hated attending hockey games; she would begin shivering in the stands within minutes.”
Second, a semi-colon may be used within a list to separate items that already use commas.
For example, “While visiting colleges, my parents and I visited Cambridge, Massachusetts; Princeton, New Jersey; and Philadelphia, Pennsylvania.” Often, when used in this way, the semi-colon is informally called a “super-comma.”
For more information on correct grammar, be sure to visit Purdue University’s Online Writing Lab (OWL).