Bananas, despite their ubiquity and general inoffensiveness, are actually pretty strange. To begin with, the banana tree is not a "tree." It's actually a big-ass herb, which makes the banana fruit a big-ass berry of sorts. The sweet, soft, seedless banana cultivar that is piled high in our supermarkets is the Cavendish, and pretty much every one of these bananas is a genetic clone. It's named after a 19th-century British aristocrat who had a thing for horticulture and Whig politics.
When we say "banana" we generally are talking about the Cavendish, but there's a whole range of banana types -- red bananas, saba bananas, Ugandan beer bananas, and "cooking bananas," commonly referred to as plantains.
For a person raised exclusively on Cavendish bananas, the plantain is a deceitful bastard. You can't just peel and eat a yellow plantain as you would a Cavendish. First off, the woodier peel makes it pretty difficult to peel by hand. And once you get it open, you'll find that they have a much higher starch content (and lower sugar content) than the Cavendish. That starch-sugar balance is why a yellow plantain isn't "ripe" like a yellow Cavendish -- as a banana ripens, the starch breaks down into sugar, and it becomes sweeter and softer. The higher starch content means you'll want to wait until the plantain is good and brown. Then you can get to frying.
Put a large quantity of canola oil into a pot or enameled Dutch oven (anything with high sides) and set it over a medium-high flame. You want the oil to be good and hot, like 350 or 375 degrees, before you start throwing plantains in.
Cut the peel off the plantain and slice it on the bias so you have angled pieces of plantain instead of rounds. Why slice on the bias? GOOD QUESTION BRO! Two reasons: first, it looks better; second, you're deep frying, and when you deep fry you want to maximize the surface area of your pieces to get as much caramelization as possible. The frying will bring out more of the sweetness and give a little crisp on those sharp-angled edges (which are there because you sliced on the bias like a pro).
When the oil is hot, carefully drop the plantain slices in and fry them for about 90 seconds or two minutes or for however long it takes them to achieve a good, deep golden brown color. Fish them out with a slotted spoon or spider and transfer them to a paper towel-lined plate to drain off the excess oil. Serve them while still hot.