When should you use a semicolon?

to separate two independent clauses that are not joined by a conjunction

• Example:
  The participants in the first study were paid; those in the second were unpaid.

to separate elements in a series that already contain commas

Examples:

• The color order was red, yellow, blue; blue, yellow, red; or yellow, red, blue.
• (Davis & Hueter, 2006; Pettigrew, 2003)
• age, M = 34.5 years, 95% CI [29.4, 39.6]; years of education, M = 10.4 [8.7, 12.1]; and weekly income, M = 612 [522, 702].

(adapted from the sixth edition of the APA Publication Manual, © 2010)