Use semicolons as you would periods, but do not capitalize the second clause. There must be an independent clause (which is, essentially, a complete sentence) both before and after a semicolon.
Do not separate a dependent clause from the main clause with a semicolon; doing so turns the dependent clause into a fragment. See FRAG.
Semicolons are also used to separate items in a list when the items have internal punctuation:
While the catalogue of the 1623 Folio divides Shakespeare’s plays into three categories – Comedies, Histories, and Tragedies – scholars often deploy finer distinctions: romantic comedies, such as As You Like It; dark comedies and problem plays, like Measure for Measure and Troilus and Cressida; and tragicomedies or romances which include The Winter’s Tale, The Tempest, and Henry VIII.