If you love southern food, especially Cajun or Creole like I do, you've already acquainted yourself with The Holy Trinity. A combination of green bell pepper, celery, and onion, this combination is the foundation for all Creole and Cajun foods. You'll find it from gumbos and jambalaya to pastry fillings. I think the only place it isn't found is in desserts. The trinity is very similar to other food foundations, such as the French miripoix (onions, carrots, celery) or the sofrito of Spanish cuisine (garlic, onion, tomato). They all operate the same way -- chopped very small and sauteed in fat (oil or butter) until very well softened, then other ingredients added on top with a liquid usually, then braised so their flavors can incorporated throughout the dish.
But of course, even this simple combination can vary among those claiming to be "purists"....
Let's first talk about the pepper. With so many kinds of peppers out there, which one do you use? Does it matter? Yes, it does. First, for a true Trinity, you want a bell pepper. Usually this means a green bell pepper, but sometimes if making certain dishes you can use a red bell pepper. Is there a difference? Yes -- the red bell pepper is sweeter than the green one so if you're making a really savory dish like gumbo you might want to use green if you want to keep it on the savory side, or use red to balance it out. You also want to think about the color of your dish -- if working with shrimp like for say shrimp creole or shrimp pockets, you may want to go with red if you want to keep the color of the dish more like the shrimp's color; or go with green if you wanted a slight color contrast. Either way, make sure it's a bell pepper. When you start using other peppers like the slightly spicy poblano, you're entering now Spanish cuisine territory. Which is fine too, but I personally like to keep it authentic as possible.
Most often the celery part is just that -- celery. However, I've seen some people use celery root. I guess you can, but I don't know why you necessarily would unless the recipe specifically called out for it. Celery root is going to give the dish a different color and sweeter flavor, so unless you want those characteristics on purpose, then stick with good old fashioned regular green celery.
And the onion. There are sweet onions like Vidalia or Maui and then savory onions like white or brown onions. I like to go with a regular, big white onion. Using a sweeter onion like Vidalia or Maui will give a considerably sweeter Trinity, and overall sweetness to your entire dish. Using white or brown onions will keep it on the savory side. When making foods like gumbo or jambalaya, I like to use savory bases because it makes the entire dish more grounded in my opinion. Especially when adding other ingredients like shrimp or okra that are sweeter, I like those flavors to come out as sweet; if the base is already sweet because I used Vidalia onion and celery root, then I won't be able to taste the shrimp as well as I could if they were contrasted with a savory base!
I've also heard some cooks say that any onion can count as the onion portion. This means that leeks, shallots, scallions, and even garlic can satisfy the onion component. I believe this is categorically false. There is a HUGE difference between using garlic and onions in a gumbo for example. You need the onions in their amount in large part to help build the body of the entire dish as well; to use that amount of garlic to provide the body would be insanity. Same thing if you used leeks or shallots -- they would simply disappear in the dish and leave zero texture. That said, if you were making a different dish like shrimp pockets you could use a milder leek to help the shrimp flavor stand out more. So pay attention to what the recipe says: if it says white onion then use white onion -- not Vidalia or Maui, not leeks, not garlic instead -- it's there not only for flavor but also for body and texture of the dish!
So how about how much of all of this do we use? The standard ratio seems to be 1:2:3, meaning 1 part of celery, twice as much of pepper than celery, and 3 times as much of onion than celery. Other sources say it's all equal parts, 1:1:1. While my personal opinion, based on my own cooking, I love 1:1:2, meaning equal parts of pepper and celery and double that mount of onion. So, for example, for a typical gumbo I'd do 1 cup of bell pepper chopped small, 1 cup of celery chopped small, and 2 cups of white onion chopped small.
There are a few other ingredients in Creole and Cajun cooking that are found in the dish 99.9% of the time, but are not included in The Trinity. Playing off of the Trinity name, here are my names for them:
Mother Mary: cayenne pepper
Saint Peter: garlic
God the Father: roux
These of course are not real terms, but my own. Feel free to use them also.
Now that you have been blessed with the Holy Trinity, get cracking on that gumbo!