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