A damage hitbox is one kind of hitbox.
When an attack hitbox makes contact with one’s damage hitbox, he takes an attack.
However, if one’s own attack hitbox is laid on top of that attack hitbox, then it becomes a matter of priority instead (see stuffing).
This does not always fit with how it looks like it will be, so caution must be exercised.
In most 2D fighting games, this hitbox exists neither during wakeup nor while being blown away; thus, one cannot take an attack during these brief periods, but this is a different idea compared to the invincible hitbox, which allows one to not be attacked in much the same way.