The question above shows some of the confusion with calling this a "dropped third strike". That term leads many to believe that the pitch must actually hit the catcher's mitt, then fall out, for this rule to be in effect.
While the term "dropped third strike" has become commonly used, and most people know what you're talking about when you use it, this isn't really the correct terminology.
Most rule books will refer to this as an "uncaught third strike". This correct term satisfies the definition of a pitched ball being caught "in flight" (that is, without touching the ground) and covers two aspects of this rule that cause the most confusion.
The first is the pitch that passes the catcher "in flight", then goes right past the catcher without being touched. That pitch is considered to be "uncaught".
The other is a pitch that bounces off the ground then is gloved by the catcher. Since this pitch did not reach the catcher "in flight", it is also considered to be "uncaught".
When the conditions for an uncaught third strike are in place (number or outs, first base occupied, etc.), the batter can try for first on either of these two pitches, just the same as if the pitch hit the catchers mitt and was dropped.