A God of the gaps argument is one that argues that since some phenomenon is unexplained, it must be due to God. It is also a form of non sequitur, since the hand of God is posited without proof and often with complete disregard to other possible explanations.
Sometimes a subject such as evolution is not understood by the speaker but may be well understood by many others, such as scientists. Of course, evolution is not a theory of chance, and has well established mechanisms underlying it.
For Bill O'Reilly: The moon causes the tides, due to gravitational tidal effects as it revolves around the earth.
Even when a subject is not well understood (i.e., the origin of the universe), that is not sufficient grounds for assuming an unproven answer like "God did it". Since the "explanation" of God is more complex than the entities that are purportedly explained by God, introducing God without evidence is simply begging the question.
There is a time where people need to understand that there are certain things that we currently do not possess the technology to know about. This is where the dreaded truth must come in - I don't know.
- "Scientists can't explain how life came to be. There must have been a god to create the first life form."
- "The Big Bang theory doesn't explain what caused the Big Bang. There must have been a god to set the universe in motion."
- "The bacterial flagellum is too complex to have evolved through natural means. Therefore, an intelligent designer must have been involved in its formation."
- "Even if the theory of evolution is correct, it doesn't explain how the first life form arose. Perhaps God's hand created life and set evolution in motion."
- "Scientists can't explain everything about how consciousness arises, therefore something divine must be at work in conscious beings."
- "Tide goes in, tide goes out. Never a miscommunication. You can’t explain that. You can’t explain why the tide goes in." 
The argument from ignorance is, at heart, an Enthymeme, a syllogism with an unstated premise:
- I don't understand how x could have happened.
- Anything I don't understand is caused by God.
- Therefore, God caused x.
(unstated premise highlighted.)
Gaps are shrinking
A god of the gaps argument is an argument from ignorance: it boils down to "We do not know how X happened, therefore X was caused by a god." However, ignorance is never an argument for something. It merely means we do not (yet) know the cause of the phenomenon.
To see why this argument is a fallacy, we can consider similar arguments could have been made at different points in human history:
- 2000 years ago: "We do not know what causes lightning, therefore it must be a god throwing lightning bolts from the sky."
- 1000 years ago: "We do not know what keeps the planets in their courses. There must be angels pushing them along."
- 500 years ago: "We do not know what causes diseases, therefore they must be punishments from God."
- 200 years ago: "We do not know how the many species of plants and animals could have appeared, therefore God must have created them."
- 100 years ago: "We do not know how the universe started, therefore God must have done it."
- 60 years ago: "We do not know how genes are passed from parent to child, therefore traits must be imprinted upon the soul."
As new explanations emerge, the gaps in our knowledge shrink, leaving less and less room in which to fit a god. Since human knowledge keeps growing all the time, it does not seem like a safe bet to assume that any given gap will remain one for very long.
An insufficient explanation
Another objection can be made to the argument's means of ignoring the question it originally intends to answer. For example, answering "What caused the big bang?" with "God did it" still does not answer the question of origins, as the god inserted into the gap still requires an explanation.
How, not What
Theists are frequently intolerant of scientific concepts that seek to provide naturalistic explanations. It is not difficult to reach a "compromise" where the theist adopts the full scientific explanation without challenge. By asking the theist "How did God do this?", the theist generally becomes receptive to the scientific explanation.
By presenting arguments in a manner that theists can accept, they gain knowledge, which is always poisonous to theistic belief.
The real question isn't "Is it possible that God exists in the unknown?" it's "Is it probable?" We should be concerned with whether or not a thing is actually true or likely true - not whether it's possibly true.
A Leap of Faith
Even if there is some supernatural being behind what science can't explain, what proof is there that it is the God of Classical Theism rather than Zeus, or Amun Ra, or Cthulhu?