The rules below are based on the Chicago Manual of Style (CMOS).  Please note that there are slight variations and preferences when using other style guides.

Independent Clauses

When a sentence contains two independent clauses, you can join them with a comma and coordinating conjunction (see Comma Guide), or you can use a semicolon in one of the three manners below.

No Connecting Word or Phrase

Conjunctive adverb:
However, therefore, besides, nevertheless, and the like

Transitional Phrase:
For example, for instance, as a matter of fact, in other words, and similar phrases

Between items in a series

When items in a series contain internal punctuation, a writer may use semicolons to separate the items for clarity.

Remember, you cannot use a semicolon to connect two grammatically unequal parts of a sentence.