Easy to explain, and I will do it for both revolvers and pistols. Also, there is a type of handgun that is also known as Double Action Only.
Single Action -- the hammer must be manually cocked every time the pistol is fired.
Double action -- the hammer may be cocked prior to firing, but the pistol can be fired by pulling the trigger alone. Trigger pull is almost always much lighter if manually cocked first.
Double action only -- hammerless revolvers (e.g. Smith & Wesson J frames) are always in a state of "hammer down" and will have a heavier trigger pull to engage the firing mechanism.
Pistol (all pistols must have a round placed into battery before firing, which in most cases cocks the hammer).
Single Action -- After a round is in battery, if the hammer is dropped for any reason it must be recocked for the weapon to fire. The 1911 is a perfect example. The manufacturers recommended carry is cocked and "locked" (safety on).
Double Action -- After a round is in battery, the weapon may be fired as with a single action pistol, or the hammer (or striker) decocked. This is usually done with a lever (e.g. Sig Sauer, Beretta, Walther PP series) or button on the pistol (e.g. Walther P99). A block usually prevents accidental discharge in the case of a drop. These pistols may be fired with the hammer down or striker decocked, but the trigger pull will be heavier due to the extra effort of recocking the firing mechanism.
Double Action Only -- Like the aforementioned revolver a LDO pistol will always be in an uncocked state, regardless of whether or not a round is in battery. In some cases the trigger pull is still very light (e.g. Para Ordnance LDA) but many pistols will have a heavier trigger pull (e.g. Colt Double Eagle, Ruger LCP).