So I thought I'd take a stab at explaining it here. You know you've learned something when you're able to turn around and explain it coherently to another person. For anyone that's new to the Myers Briggs Personality Type, this might be foreign territory and it might help to read my post on the basics first. And if this is old hat for you, I'd love to have you tell me where I've got it wrong.
Determining Your Dominant Function
I'm not sure who first came up with the dominant, auxiliary, tertiary and inferior function idea, but I think it was Carl Jung (who came up with a lot of the foundational ideas that the Myers Briggs Type Indicator is based off of). The basic premise is that there's one mode of operation that we're best at and that we use most often. (We're probably best at it precisely because we use it most often.) This is our dominant function. We tend to feel most comfortable and certain of ourselves when we our using our dominant function. There are times, however, when a different tactic is called for, and when that happens, we often revert to using our auxiliary function. The auxiliary can also be used to support the dominant function. (So rather than being an "instead of" mode of operating, it becomes more of an "I will approach the task using the auxiliary so that I can better user my dominant function with the results of my use of the auxiliary. ... Of course, we don't think out our actions in these terms, but it still describes the method that we often use to function, even if we don't realize that's what we're doing. Remember, the MBTI is a tool that was developed to describe how things already are, not to force us into some new means of interaction.)
Once you've determined which of the 16 possible type combinations you fall under, you can figure out what your dominant function is by first looking at the last letter of your psychological type. Are you judging or perceiving? If you are judging, then look next at that third letter, which indicates the judging function that you prefer. Are you thinking of feeling? Likewise, if you are perceiving, then look at the perceiving functions, which are sensing and intuition. If you are an extravert (look at the first letter in your type) then your dominant function is the judging or perceiving function that you just came up with. And your dominant function is written as one of the following:
Se Ne Te Fe (the little "e" indicates the attitude of your dominant function)
If you are an introvert, then you have one more step. If you ended up with a judging function (T or F), then your dominant function is your perceiving function (either S or N). If you ended up with a perceiving function then your dominant function is your judging function. Your dominant function is written as one of the following:
Si Ni Ti Fi (the little "i" indicates the attitude of your dominant function)
Let me give two specific examples to help make sense of this. (It took me a couple of readings to wrap my head around this. I don't know why it seemed so complicated, but once I figured it out it seems pretty straight forward.) Let's start with an ENFP. Because this type is perceiving, then we want to look at the perceiving function (in this case N) to determine the dominant function. When this person is "in the world" (or extraverting) then this person's dominant function is intuition which is written as Ne (Intuitive with an extraverted attitude.) If we're looking at an introvert, however, there will be an extra step. I'll use myself as an example. I am an INTJ. Because I am judging (That doesn't mean I'm judgmental, though I am at times.) I want to first look to my judging function, which is thinking. But because I'm an introvert, my dominant function isn't what I do when I'm extraverting, but what I use when I'm introverting (which is where I prefer to be most often). So instead of having thinking as my dominant function, I look to my perceiving function, which is intuition. My dominant function is intuition with an introverted attitude, which is written as Ni.
The Auxiliary Function
The auxiliary function is easy to figure out. If your dominant function is judging (T or F) then your auxiliary function is perceiving (S or N) and vice versa. And if the attitude of your dominant function is extraverted, then your auxiliary is introverted (and vice versa.) So if you are an ENFP and your dominant function is Ne, then your auxiliary function is Fi. If you are an INTJ and your dominant function is Ni, then your auxiliary function is Ti.
The tertiary function is one that you only fall back upon if your dominant and auxiliary aren't working for you. Because you don't use it as often, you're not as good at it. So, for example, if your tertiary function involves intuition, then you'll use intuition, but you won't use it nearly as well as someone who has that as a dominant function. Quenk gives some great examples of strong and weak uses of various functions. I'll just give one quick example here so you get a better sense of what I'm talking about. A dominant thinking person might be very good at giving impersonal criticism. A tertiary (or even more so, an inferior) thinking person will more likely give excessive criticism.
To figure out your tertiary function, take a gander at your auxiliary function and select it's oppose. So if your auxiliary is Si or Se, your tertiary will be N. (There's apparently debate about whether tertiary functions have attitudes. I don't get any of that bit yet so I'll just shake my head a bit here and move on.) If your auxiliary is Fi or Fe, then your tertiary function is T.
Here's the part that I had been most interested in when starting Quenk's book. I had been reading Lynne Baab's book, Personality Type in Congregations, and she mentioned the inferior function somewhere in there. I probably wouldn't have thought much of it, but there was a description that jumped out at me. It was similar to the one I gave above of someone who is strong as a thinker verses someone who is using that function but isn't very skilled at it. But it described my son and much of his recent behavior. I had just the week before been thinking to myself, "Nathan has behaving so differently lately. It's like I don't know my own son. I never would have expected him to think this way or act this way. It's unlike the kid that I've known for over a decade now. Is it just because he's a teen?" Then I saw the description in Baab's book and kathunk, it all slid into place. Nathan has been under a lot of stress lately (not just from friends and school, but because of health issues as well) and he's been reverting to his inferior function, which is exactly the opposite of his usual behavior.
To determine your inferior function, look at your dominant function and reverse it. So if your dominant function is Ni, then your inferior function is Se. If your dominant is Ti, then your inferior is Fe. And because this is not just something we don't use often, but that is even the exact opposite of what we are really good at, we tend to do it very poorly. In a situation where a person is dominant extraverted feeling (Fe) they will often have a sensitivity to the welfare of others. But if someone has Fe as their inferior function, they will likely have a hypersensitivity to relationships when they are under severe stress.
We use our four functions in various settings. Oftentimes, if we try to use our dominant function and that doesn't seem to get us anywhere, we'll revert to our auxiliary function and on down the line until we finally use our inferior function as a sort of "last resort" measure. We might also learn to use our other functions to serve our dominant function. (I don't really fully understand this bit yet. I've got the main idea, but how it plays out is still fuzzy in my mind.) We might not attend to details as much in our dominant function which is thinking or feeling, but if our judging function needs some supporting evidence, then we might use our auxiliary or tertiary function to help gather the details that our dominant function needs to work well.
As we grow older, we often naturally begin to explore our inferior function more. So someone who has grown up being very logical may explore their feeling side more. (This is something that has come up several times in the books on personality type and spirituality that I've read. We may have grown up feeling much more comfortable in one type of worship over any other type. But as we reach middle age we might check out a congregation with a very different worship style. This exploration often helps us to feel closer to God as we stretch and grow in what was before a very uncomfortable area.) I suspect this is why we have a stereotype that when people hit middle age they begin behaving strangely. What's happening is that people start exploring their inferior function, but since they haven't used it much in their first 3 or 4 decades, they're not really so good at it. So they come across as being bumbling fools hitting a middle aged crisis. That's not all bad. It can be a chance for a person to find new interests and skills that will help them to grow in areas that they'd previously avoided.
I have about a quarter of Quenk's book left to read. I still feel very uncertain of what each function actually looks like when it's being acted upon. I get that Se means somehow interacting with details in the outer world, but how does that play out? I've picked up a few ideas (an Se will be more open to exploring new tastes or sights or textures) but if you throw a function at me, it would take me awhile to sort out how that looks. (What does it look like to be an Si? You tend to be more interested in stuff inside yourself, but what? Not tastes or sights. Ideas? Thoughts? Facts?) I'd love a list of examples of how each function plays out. That's probably something that I'll explore next.