Here is a more recent recap of executive functioning that does a good job giving examples of how deficits in executive function affect our ability to perform various tasks.
I would say that executive function is the ability of our brain to choose which pieces of information belong to a puzzle, then puts all the pieces of the puzzle together and uses that "big" picture to make decisions. This ability is theorized to exist or be controlled by the frontal lobes - the most recently developed (in terms of evolution) part of our brain.
Executive function allows us to
- sort the information coming in (the teacher's notes on the chalk board go with the lecture but the smell of food from the cafeteria does not)
- assemble the information into a complete picture (the teacher is talking about how to do addition and the notes on the board are examples) and
- decide how important it is to be able to remember the information or use the skill again in comparison to all the other things our brain needs to be able to consciously recall at will (addition is what I use when I need to know how much money I have and that's important so addition is important and I need to remember this skill).
This can happen on the fly (deciding right now which parts of a lecture or story to pay attention to or are important and need to be remembered based on the sum of our previous experiences and knowledge) - or in a more reflective state (thinking ahead to what the family schedule is over the next few days and deciding to go to the store today and what to buy because you foresee that there's no time to do it later in the week).
In ADHD deficits in executive function are theorized to occur because your brain has trouble suppressing some kinds of input (often external stimuli) in favor of other input (often internal processing of information).
For example, a child with ADHD who sits near the window may have trouble concentrating on the teacher's lecture because his brain is biased in favor of attending to external stimuli that could signal danger. This interferes with his brain's ability to attend to external stimuli that are highly unlikely to signal danger. His brain literally can't choose to attend to the teacher's lecture because it is focused on the noises coming through the window (the birds singing, the lawn being mowed) that could be signs that danger is approaching.
One theory about why this happens has to do with the evolution of the brain. The "older" parts of the brain react to our environment in a more primitive way. The sounds, smells and sights we perceive that signal potential danger or reward such as food are powerful and automatic. Keeping us alive is the priority. You can't pay attention to a lecture if you smell smoke. You have to stop and figure out where the smoke is coming from because smelling smoke triggers your "old" brain to scream for you to pay attention to this potential threat.
So that part of our brain is competing for our attention with the part of our brain that is trying to learn how to read.
The ability to selectively attend to information based on importance to our long term well being is "fragile" in that it is more easily disrupted in some people.
Most people are able to ignore the fact that they hear a large, noisy machine outside (the lawn mower) because they are able to classify that input as non-threatening and therefore less important than the lecture.
Other people simply cannot do this because they can't override the part of their brain that thinks this noise signals the approach of something threatening.
Other things that can disrupt executive function include -
strong negative emotions
lack of sleep
too much information coming in all at once
pain