One of the most common terms applied to the Father is "Almighty" (twice in the Apostle's Creed alone). On the surface, it's a simple concept: God can do anything. When we start thinking of the logical consequences of omnipotence, however, we find ourselves in a relatively famous philosophical quagmire. God is presented as omnipotent (all-powerful), omniscient (all-knowing), and omnibenevolent (all-loving). The existence of evil would seem to indicate, however, that God cannot be all three. If God were all-loving, it should follow that He would want to prevent any evil or suffering from occurring. That evil still occurs, then, would indicate that either God is not aware of all the evil, or He is powerless to stop it. Conversely, if God is aware of all evil, and has the power to stop it, but chooses not to, how can He be considered all-loving?
The answers to these questions are not found in the Bible. The Bible only says that He is all these things and doesn't really make much of an effort to reconcile the three. Indeed, the Bible is full of examples of God apparently inflicting a lot of pain, death, and suffering on people for reasons that are not always made clear. Perhaps the most extreme example is Job. God allows Satan to take Job's life apart, for apparently no better reason than Satan dares Him to. But, it is also in Job's story that we get a hint of how to move past the contradictions.
Job, while not blaming God for His troubles, requests an explanation of what he has done to deserve them. God does not answer Job's question, but instead makes it clear that Job does not have the right to be answered, and strongly implies that Job wouldn't understand the answer even if he were given one. But here is where the seeming contradictions between God's omnipotence, omniscience, and omnibenevolence start to fall apart.
It seems clear that we don't really understand the terms of the argument. In fact, it seems doubtful that we could ever understand them completely. Certainly, we have glimpses, employing similes and metaphors, extrapolating from the understanding of our own power, knowledge, and compassion. But, not being omniscient ourselves, how can we define and account for all of the actual consequences of omniscience? Similarly, we must also be limited in our understanding of how omnipotence and omnibenevolence can be expressed in our finite universe.
This, of course, is very intellectually unsatisfying. We're not comfortable with realities that we can't stick in a well-defined box. So, we'll pretend that we do understand the terms. However, it would also see that we cannot define them so narrowly as the original statement does. In particular, it is not helpful to define "omnipotence" as the "ability to do anything that can be done".
In fact, there are many things we know that God cannot do, because they are nonsensical and are contradictory to logic. It is not possible for God to make a pencil that is not a pencil. It is not possible for God to create a rock so heavy he can't lift it. This does not mean He is not omnipotent. We also know that God cannot perform actions contrary to the nature of His perfection. He cannot lie, or be cruel, or sin, or even learn (being omniscient). This is not due to lack of power or ability, but due to the fact that He does not lack perfection.
With these considerations in mind, we may better think of "omnipotence" as having power and authority over all of creation. This leads to the following line of argumentation to address the "Problem of Evil":
This argument should seem somewhat familiar to those with even a small amount of Biblical knowledge. We are given many assurances that suffering can actually be good for us. Every mature person, religious or not, can understand that some negative circumstances now can have great benefits later ("No pain, no gain"). Every good parent knows that they must purposely inflict or allow some amount of suffering on the part of their children. Parents should not be pleased that this occurs, but can recognize that it retards spoilage, and is necessary to achieve the best final outcome - another mature human being. Similarly, it is possible to trust that God truly does know what He is doing and that someday, though present evidence may seem to indicate otherwise, this will be the best of all possible worlds.