This question has been around at least since Plato, and is known by the name of the "Euthyphro Dilemma". It's really nothing more than a silly trick question wearing a "deep" philosophical mask.
If one were to go with the first option provided, it leads to the conclusion that God is subject to the authority of Objective Morality and is not actually able to do whatever He wants, making Him not really God. In other words, God has rules that He has to play by, which means He is not in charge.
If one were to go with the second option provided, then it leads to the conclusion that God is able to command whatever He wants, and that we call this stuff "good" just because God commanded it, making "good" entirely subjective. "Good" could mean one thing one day and something totally different the next, depending upon the random whim of God.
Generally, the non-theist uses this question to attempt to demonstrate that either God does not exist (option 1), or else Objective Morality does not exist (option 2), claiming that they are incompatible.
We can see the flaws in this claim better, I think, if we look at it as a simple mathematical statement.
Is the authority of Objective Morality (AOM) "greater than" the authority of God (AG) (option 1), or is the authority of Objective Morality (AOM) "less than" the authority of God (AG) (option 2)?
Is AOM > AG or AOM < AG?
Those are the two options provided.
Now, anyone who has ever put together a math statement using "greater than" or "less than" symbols knows full well that they are not the only symbols that can go in the middle of the statement. There is also "equals". This would give us:
AOM = AG
...which is, of course, the only possible logical conclusion, and is also the Christian position.
Let's look at this in a little more detail.
Generally, this dilemma is posed in response to the Moral Argument for the Existence of God. We won't get thoroughly into the Moral Argument here, it's validity and objections to it, or responses to the objections, etc, but we'll present it here just for the sake of background. The Moral Argument basically goes like this:
a. If there is no God, there are no objective moral values.
b. There are objective moral values.
c. Therefore, there is a God.
This argument is really the same for any kind of prescriptive "oughts", including not only morality, but justice and logic as well. We ought to behave morally, we ought to judge in a just manner, and we ought to think logically. Without the Christian God, there is no "ought" for anything.
Now, either (A) these laws are defined by the character of God and are inseperable from Him, working in eternal conjunction with Him (the Christian assertion), or else (B) they are not. If they are not, they are either (1) subject to Him, (2) equal in authority to Him and occasionally opposed to Him, or else (3) reigning over Him.
(B1) They cannot be subject to Him and yet not reflect His own character, because if so, they would not be authoritatively objective, and yet they are. If they are subject to Him and reflect His own unchanging character, it is the same as saying that they are defined by the character of God and are inseperable from Him, which is assertion (A), the Christian assertion.
(B2) Since God, the beginner of time, caused all things that ever have been and ever will be, these laws cannot logically be equal to Him and occasionally opposed to Him, or else their opposition to Him would be caused by Him, making them subject to Him (B1), rather than equal to Him.
(B3) They cannot be in authority over Him because then He would not be God, and He is.
Thus we see that (A) is necessarily true because of the impossibility of the contrary, and further, we know the character of God by the objective authority of the "oughts" He has given us. God is Good, God is Just, and God is Logical.
Another way to phrase the Euthyphro Dilemma is to ask, "Should we obey Objective Morality, or should we obey God?"
And of course, the answer to that question is...