What are the functions of vinegar in cooking?

Vinegar is an acid. It is made from fermenting ethanol. This makes acetic acid.

Much like a kid's vinegar + baking soda = volcano science project, vinegar is sometimes used in breads to help the bread rise. My mom's banana bread is one example. And the more ripe the bananas are, the less acidic they are, so you need more vinegar in her recipe. My wife's mom's Banana bread recipe uses baking powder for the same effect using different chemicals.

Note: There is a difference between baking soda and baking powder.

Because it is acidic, it can be used in marinades to break down meats (or connective tissue). Follow the recipe for the marinade.

Vinegar (since it is an acid) also speeds up coagulation.

You can also use vinegar to preserve things like cucumbers. This is the process of pickling. The acid helps preserve the cucumbers (or other vegetables or fruits). Instead of vinegar (or other acids), you can use other, more natural methods of fermentation to pickle the vegetables. I've never done it, but I hear that they spoil quicker than the vinegar style pickling.

If you want to get into the science of cooking, I recommend "Professional Cooking" by Gisslen - it's a textbook for future chefs in culinary school

I don't remember the exact science behind the potatoes not turning to mush ... I know it was discussed on an episode of America's Test Kitchen, and they even discussed the amount of vinegar and how it affected the window for cooking times. (I want to say it was on an episode where they were making an Austrian potato salad)

... and it's not specifically vinegar, and not solely potatoes. Acids will stop quite a few items from softening when cooking. I know the list inclues onions (pre-cook onions before adding acids if you want them to disolve into a sauce), potatoes, apples ... I'm guessing there's others.

For potatoes and apples, most recipes claim it's to slow down browning. You see it in potato salad, but not mashed potatoes. You'll see it in apple pies, but not in applesauce ... because they'll come out lumpy.

Matt already mentioned three uses for vingear:

• as half of a chemical leavener
• tenderizing / marinades
• preservation / picking
• coagulation (eg, when poaching eggs)

But also :

• stabalizing eggwhite foams (might just be a variation of coagulation?)
• denaturing / "cooking" (eg. in the case of ceviche; a variation of marinating?)
• flavor (might not be a 'science case' ... but sour is one of the primary tastes; especially helps to balance out fatty dishes)
• cleaning / disinfection (might not be a 'cooking' use)

You forgot that a tablespoon of vinegar in a cup of water with food coloring has been used by everyone to dye Easter eggs.

Vinegar is also used to form many emulsions, including Hollandaise sauce (with egg yolk and butter), mayonaise (with egg yolk and oil), vinaigrette (with salad oil).

it can also be used for making buttermilk, by adding a tablespoon of white vinegar to a cup of whole milk