In an auto pistol, Double Action (DA) means the pull of the trigger both cocks and releases the hammer, or striker. This necessitates a longer, often heavier pull. (Anything with DAO in the name, Glocks, and M&Ps to name a few) Like a Double Action Revolver.
In a Single Action (SA) automatic, the rearward movement of the slide either from loading (XD) or firing (1911), completely cocks the hammer or striker. Pulling the trigger only releases the sear, releasing the hammer or striker. (XDs, 1911's) Like a SA revolver, but rearward movement of the slide cocks the "hammer", not your thumb.
SA/DA designs, like most Sigs, HK, and Walther are a combination Single Action/Double Action. The gun is carried hammer-down, in double action. The first trigger pull (long and heavy) cocks the hammer, and the gun fires in single action for the remaining shots.
Advantages??? All subjective. DA is considered "safer" since the trigger pull is heavier, and longer, but if you don't intend to kill something, your finger shouldn't be on the trigger anyway. SA is considered "more accurate" since the lighter/shorter trigger pull is "less likely" to disrupt your sight picture.
SA's are most often carried "Cocked and Locked" (Already in SA mode with thesafety on). DA's are all carried "Hammer Down".
Hope this helps!