For centuries, atheists and skeptics have tried to argue against the existence of God on the basis that evil exists in a world where a good and powerful God exists.  That evil exists I have no doubt (as C.S. Lewis once mused, "Have [people] never been to a dentist?").  And it seems evident to Christian believers that God does indeed exist.  So, what then is the problem?  To see this we have to illustrate what it means for two propositions to be incompatible.  Take the following:

(a) Joe is a bachelor

and

(b) Joe is married

It is inconceivable that these two propositions should be true at the same time.  Why?  Because we know that being a bachelor just means to not be married.  Hence, the two are deemed logically impossible at the same time.  Now take the following two propositions suggested by our atheist critics:

(c) God exists

and

(d) Evil exists

Is the same force of self-inconsistency evident here as it is with our friend Joe?  It doesn't appear obviously as such (not like a "married bachelor" at any rate).  The Scottish skeptic David Hume of the 18th century once asked,

Is [God] willing to prevent evil but not able?  Then He is impotent.  Is He able, but not willing?  Then He is malevolent.  Is He both able and willing?  Whence then is evil?" (Dialogues Concerning Natural Religion)

So presumably the inconsistency has to do with God's being both omnipotent (all-powerful) and omnibenevolent (all-good) while evil remains in the actual world.  God being all-powerful we would surmise that God has the ability to abolish it (after all, Revalation suggests that this will in fact happen eventually ... why not now?).  But is it really true that God can create a world of free creatures where each person freely always does the right thing?  This isn't at all obvious.  Consider the following two possibilities that may be true about a mythical college student named "Ivan":

1. If Ivan were in situation S then, possibly, he would enroll in Philosophy 101

2. If Ivan were in situation S then, possibly, he would not enroll in Philosophy 101.

3. Under no circumstances will Ivan enroll in Philosophy 101.

4. God cannot determine a causal agent to bring about freely a specific action.

To acknowledge (4) as false would be to affirm a logical contradiction. No being, no matter the magnitude of his power within the constraints of his own nature, could make creatures freely do something. This is the free will defense.  Because human freedom is an inevitable factor in creating human beings, how is God supposed to guarantee that such creatures will always choose the good?  Like Ivan, some people may just be too stubborn and will never choose the good in every situation no matter what.  So, it may not be possible for God to construct a world where every person freely chooses the good.

Now, what about the idea that God wants to create a world where there is only good in it?  Well, it sounds true but upon closer inspection is actually questionable.  Maybe only those worlds where there is only good in it are grossly underpopulated - say they only have three people living in them.   It does not appear obvious here that God would prefer this world only because everyone is good for it is completely lacking in population.

Keep in mind, these scenarios need not be actually true, they need only be logically possible.   Remember, in order for there to be a logical inconsistency between two propositions they cannot have any imagined scenarios where the two might coexist.  But in the case of "Evil exists" and "God exists" we do have imagined scenarios: (i) God cannot create a world where everyone freely chooses the good, and (ii) God doesn't want to create such a world because they are underpopulated.  As long as (i) and (ii) are possible, the allegation that evil and God are incompatible fails.