Short Answer: When we say "God is omnipotent" we mean "He has the power to do whatever He pleases". There are other definitions, such as "capable of doing anything man can think up", but by such definitions He would not be considered "omnipotent".
As indicated in the wikipedia post you linked, there are many definitions for the word omnipotent. As with many of the heavily debated topics in Christianity today, the controversy arises when we begin to mix two different definitions during a debate. (Other examples: "evolution", "knowledge", "free-will", "love", "life", "evil", and so on.) The word omnipotent literally translates as "all-powerful", but as usual, the meaning of the word is determined by the person using it, and is clarified by the context of their usage.
So here's the story behind this controversy: Christian theologians have studied the Bible and noticed some things about God (such as His great power), and have summarized those things using simple terms (such as omnipotence). For a Christian, the word omnipotent is simply a term used to refer to the teachings of Scripture about God's great power - nothing more, nothing less. Heathens (in their blindness and ignorance) have noticed Christians attributing "omnipotence" to God, have used a different definition of the word which is based on a misunderstanding of the Christian usage, and have then used this heathen definition to attack and accuse Christians. Modern Christians (in their laziness) often accept the heathen definitions, neglect to study the topic in Scripture, but have faith that God must be omnipotent, and then try to reconcile their view of God with the heathen definition.
The solution to all of this confusion is for Christians to study the Scriptures and understand what was originally meant by the term omnipotence in Christian theology, and stop trying to train "unreasoning animals" to understand things which are impossible for them to understand, and which they are unwilling to accept. With that said, let's take a quick look at the definition commonly used by heathens.
The Heathen Definition
Heathens often use the word omnipotent to mean "capable of doing anything." Modern (naturalistic) heathens have a field day in debates using this definition, because it opens up all sorts of nonsensical attacks, such as "can God create a rock so big He cannot move it?", which is essentially the same as asking, "does God have the power to create a situation in which He is powerless?", or even more plainly, "does God have the power to not be God?" As C. S. Lewis pointed out, this is nothing more than shrewd, nonsensical wordplay.
A Scriptural Definition
Let's look at what the Bible says and see what sort of picture is actually painted in Scripture.
“I am the Alpha and the Omega,” says the Lord God, “ who is and who was and who is to come, the Almighty.” -Revelation 1:8
Passages such as this one teach us that God is "almighty". It is most accurate to think of this word as meaning "God has unlimited might, which enables Him to do anything He wants to do", rather than interpreting this to mean "it is possible for God to do anything we can think up." The following passage describes this more clearly:
our God is in the heavens; He does whatever He pleases. -Psalm 115:3
In other words, once God desires to do something, He has the power to do it, and He does it. This does not mean that God has the capacity to do whatever we can think up. It means God has the power to do whatever He wants to do. Of course "what He wants to do" is going to be consistent with His nature. As we learn more about the nature of God, we can better understand what sorts of things He would do, and can better understand how He works, how we should pray, etc. For example, God is love, God is good, God is just, and so on.
Thus, by the Christian definition, there are things God cannot do, because He cannot contradict His nature. For example, He is faithful, so He cannot be faithless;
If we are faithless, He remains faithful, for He cannot deny Himself. -2 Timothy 2:13
Another example is that since God is truth, He cannot lie:
God, who cannot lie, promised... -Titus 1:2
So by the Christian definition, God's actions are limited by His nature, while the heathen definition requires that God be capable of anything -- even things contrary to His "nature."
How to debate with heathens
Heathen: "Christians say God is omnipotent, but that's impossible. Let me illustrate with a question: Can God create a rock so big He cannot move it? If you say yes then He can't be omnipotent because He can't move it, and if you say no then He can't be omnipotent because He can't make a rock that big."
Christian: "That question doesn't really make sense. The rock you described is a nonsensical object. The size of a rock doesn't change God's ability to move it. That's like asking if God can make a ball so green that He can't roll it; the color of a ball doesn't impact God's ability to roll it anymore than the size of a rock impacts God's ability to move it."
Heathen: "My point is, if God is omnipotent then He should have the power to do anything, but does He have the power to make a contradiction? Can He create a square circle? Does He have the power to limit His power?"
Christian: "You are confused about the meaning of the word omnipotent when it is used by Christians. We use that word to refer to God's ability to do whatever He wants to do. Your example is equivalent to asking if God has the power to not be God. Not only is that nonsensical, but it would require God to act contrary to His nature, which He cannot do."
Heathen: "So God isn't really omnipotent?"
Christian: "God is almighty and does whatever He pleases. There are things He cannot do, such as lie, so if your definition of "omnipotent" includes the capacity and ability to do anything man can think up, then He would not be "omnipotent" according to your definition."
Heathen: "So God isn't really omnipotent. In that case He can't be God."
Christian: "You are using heathen definitions and heathen criteria for classifying a being as 'God' to show that it is impossible for the Christian God to exist. Not only is this arrogant and insane (since the human brain is far too small to draw such a conclusion), but your method presupposes your conclusion that there is no God. If God were real, He would be the only one qualified to explain His attributes and nature, and it would be our job to learn from Him. Christians believe He has done this in Scripture. If you wanted to see if the Christian view were internally consistent, you would have to look at Scripture, and according to Scripture, you would need to know Him in order to understand His ways. So if you are interested in knowing the truth, start by seeking God in prayer and reading Scripture. Once He begins to open up your eyes to the truth, these things will begin to make more sense to you. If you are not interested in knowing the truth, feel free to continue scoffing at God, but these arguments only 'disprove the possibility of God's existence' from a perspective which presupposes His non-existence. This is what the Bible refers to when it says 'the wisdom of this world is foolish to God'."