A theodicy is basically an explanation for why there is evil in the world. Many of the world's religions have some explanation for why evil occurs and where it came from. However, it is a problem for some religions which believe that everything that exists was created by a God who is all-powerful (omnipotent), all-knowing (omniscient), and all good (omnibenevolent).
Here's the problem: if you believe that God created all things, then did He create evil? And if He did not create evil, then how does evil exist?
A closely related problem is why God allows evil to continue. If you believe that God is omnipotent, and omnibenevolent, then how does God allow evil to occur? Perhaps God is not all good, so it is not a problem if evil occurs. Or perhaps God is not all powerful, and cannot stop evil acts from occurring.
One solution is that God allows humankind to have free will. Why does evil occur in the world? Because humans have free will. God lets humans act as they will, and that is why evil occurs. It sidesteps the problem by placing responsibility for evil with human disobedience.
Often theodicy is discussed in the context of western religions, and particularly Christianity. Some other religions, like Buddhism, would not have as much of a problem as they do not believe in an all-powerful Creator. If there is evil in the world, it is the result of human actions.