Alex Delany of Bon Appetit tells us that the different types of onion have its own special qualities -- a distinct onion personality. Each type -- white, yellow or red -- can do something better than the other two.
White Onion
These are usually the most mild of the onion varieties, so they are best used in salads and on sandwiches. While they're pretty mild on their own, you can further sweeten them by slicing one thinly and giving it an hour-long soak in cold water. The Vidalia and Maui onions are mentioned in the same category as white onions. Both varieties, though, contain a much higher sugar content and a lower sulfur content (which gives onions that sharp smell and flavor). Use them when a recipe calls for a "sweet onion" specifically; although, a white onion can be used as a decent substitute.
Red Onion
Sweetness is the red onion’s greatest strength. The sharpness of its flavor and the intensity of its smell are slightly more potent than that of the white onion, but the sugar content is much higher. That natural sweetness makes them a prime candidate for pickling. The other obvious draw to a red onion is its color. That deep red hue provides a nice variety in color that other onions don’t. When it comes to grilling, red onions should be the first choice. Cut into wedges, they char nicely on the grill, and their interior texture goes jammy, instead of mushy, like white and yellow onions tend to do.
Yellow Onion
You don’t want to eat a raw yellow onion. While the flavor of a yellow onion (also commonly called a Spanish onion) is fantastic in its own assertive way, it comes with a very high sulfur content. Cooking out that intense sharpness paves the way for the sweet onion flavor to shine, which is why it is preferred when caramelizing onions. The amount of flavor you’ll get from caramelizing yellow onions, as opposed to white, is significantly higher -- perfect for burgers.
What should we use when we’re just sautéing onions in butter or olive oil for some base flavor in sauces or pastas or fried rice? Use whatever you want. If it's just some generalized onion flavor backup you're after, it doesn’t really matter. All three onions will do the job just fine.