I know "make your own broth" is one of the hallmarks of people supposedly much too interested in food. One might use the term "foodie," or even "gourmand." Specifically, one other than I might use those words, as I am neither foodie nor gourmand, and yet make my own broth on a regular basis.
Vegetable broth is stupidly simple, requires little to no effort, and is made from trash. It also has the added benefit of making food taste better.
How to make vegetable brothThe first step is to establish a stockpile. Pun totally intended, BTW.
When you are chopping up vegetables for any reason whatsoever, you will generally end up with a pile of trimmings. Normally these trimmings would go in the garbage, the compost pile, your pet rabbit's bowl--wherever. To establish a stockpile, you instead put the clean trimmings in a container and put it in the freezer. As you make more food, the stockpile grows and grows. If a viable broth vegetable starts to wilt and die in your refrigerator, chuck it into the stockpile as well. After a week or two--assuming you actually cook regularly--you'll have enough (probably far more than enough) scraps for a small pan of stock, perfectly preserved and ready.
It's also totally fine to chop up whole fresh vegetables for broth, but I tend not to, because I am cheap.
Not all vegetables are great for stock. However, this is no problem: just don't save those particular trimmings. Problem vegetables include anything in the brassica family (i.e. broccoli, cabbage, kohlrabi, brussels sprouts, kale), heavily flavored vegetables, bell or hot peppers, and dill stems. The cabbage family reeks when boiled, so stay far away. Also, seriously, do not use dill stems unless you want to end up with a pan of pickle brine. You can definitely make broth including some heavily flavored vegetables if you want to; it's just that you aren't going to want beet broth in something like a pan of sauce velouté, so it's not worth keeping beet peels around unless you make a lot of beet soup.
In contrast, onions (including green onions, shallots, and clean onionskins), garlic, mushrooms, tomatoes, carrots, celery, potatoes, lighter greens such as spinach and chard, bay leaves, and parsley are all great. I personally tend to collect a lot of chard stems, carrot and potato peels, and various onion matter. I also recommend keeping every single scrap of summer tomato flesh you can. Err toward easily adaptable, semi-neutral flavors.
Once you have about two cups of vegetable bits and pieces, it's time for embrothening.
Get out a pot. I usually use a 1.5 or 2-quart pot for broth, since I almost always need my big pot for whatever else I'm cooking. You can use whatever size pot you like.
Stuff your pot with a nice varied assortment of frozen vegetable scraps. You don't need to defrost them; their liquid will all go straight into the broth. Make sure you at least have some onion or garlic in there. I occasionally add a split green onion or a couple cloves of crushed garlic if I don't have enough oniony scraps. You'll find that two cups of vegetables can nearly fill a pot this size, leaving only an inch or so of clearance. This is actually good. More vegetables = more flavor, and it all would've been trash otherwise anyway, right? You aren't wasting it; you're using it. Just cook the broth in a bigger pan if there's actual danger of overflow.
Fill your pan with water to just over the vegetable line. You'll be surprised how much water you can get in there, even if it looks like it's totally filled with vegetables already. My pot usually looks something like this:
Now bring that pot to a boil, lower the heat, cover, and simmer for about twenty minutes. That is it. Your broth is cooked.
Next step: straining.
This bit is obvious, but I want to say it anyway, because sometimes it's just too easy to let your brain turn off at the end of the day. KEEP THE LIQUID; DISCARD THE VEGETABLES. Do not accidentally pour your broth down the sink, ok?
A few straining methods work well. First, you can strain the traditional way: pour your broth through a strainer into another pot. Alternately, cover the pan mostly with its lid and pour the liquid out through the crack. You could also use a slotted spoon or mesh skimmer to lift the vegetables out. If you happen to have a pasta insert for your pot, you could cook the broth in it and just lift it out afterward. I also occasionally lower a pyrex measuring cup into my pan with the mouth held against the side; this way, liquid gets into the cup but vegetables do not. This last method is most useful when you want to keep cooking the rest of your broth for whatever reason.
For any of these methods, make sure to press the leftover vegetables after draining. Use the back of a spoon or something similar. This will squeeze out all the liquids to give you more broth.
Your broth is now ready for any possible use.
Questions?- What if you don't have enough vegetables to make the amount of broth you need?
First, it is totally fine to chop and use fresh vegetables instead of frozen.
Next, make as much broth as you can with the vegetables you have. When the broth is done cooking, strain out maybe half the liquid (which you can then either set aside or put directly into your soup/whatever). Then fill up the pot with more water and let it simmer again. I can usually do two water additions before the broth quality suffers even slightly.
What not to do if you can help it: use a boullion cube or any storebought concentrated stock mix. They are super-salty and often taste tinny or chemical. If you're desperate, use water to make up for your broth deficiency.
- What if I wanted to make, say, broccoli soup--can I use broccoli in the broth?
I wouldn't. Anything in the brassica family smells just awful when boiled too long. It would be better to make a standard broth while you cook your soup base (e.g. onion/carrot/celery in olive oil), add the finished broth to the base, add the broccoli, and simmer until cooked through, about three to five minutes.
- What do you do with broth?
You can cook all kinds of things with broth, although soup is the most obvious answer. Practically any kind of soup is good: chili, tomato, various chowders, potato leek, butternut squash. Or add the broth to softened onion/garlic/various aromatics to make a more liquid soup, and then add cooked pasta, meatballs, or dumplings.
Vegetable broth is a great braising liquid, especially when enhanced with white wine or dry vermouth. Rice, quinoa, barley, and various other grains taste excellent when cooked in strong veg broth. You can also use it to make sauces requiring water; I use mine in enchilada sauce on a regular basis. Finally, you can reduce it to make a glaze or use it in a marinade.
- No, I mean what do you do with the leftover broth?
Pour it into containers, let it cool, and put it in the freezer. It's important to let hot broth cool before you lid and chill it; if you don't, you get fermented broth. Just let your containers sit on the counter for an hour or two before you put them in the freezer.
- What do you do with the frozen broth?
You use it the same way you use liquid broth: put it in soup, grains, sauces, or what have you. I usually run hot water over the back of each container, pop out the frozen broth, and put it directly into whatever I'm cooking. That way it just melts into the soup/whatever. This does depend on what I'm cooking, however. For a risotto, for instance, you need to heat your broth to a boil before using--so I just put the frozen broth in a pan, add a little water, and heat it up.
- How long does vegetable broth keep?
I wouldn't leave it in the refrigerator longer than a week. However, if you freeze your broth, it'll be totally fine for months.
- Do I seriously only need to cook it for twenty minutes?
In my experience, that's plenty of time to make a dark-colored, flavorful vegetable broth, and you can get away with even less time if you're pressed. So: yes. I often put on a pot of broth, start cooking dinner, and use said broth in the dinner fifteen minutes later. You can leave the pot on to simmer a bit longer if you don't need it quite yet, but you aren't trying to extract marrow from bones here, so the long cooking normally associated with broth is just not necessary.
BOOM.
How'd I do? More questions?
You guys, I think the pizza party for christmas eve was the best idea ever. We are definitely going to do this as close to every year as possible. YES.
Dough: I used a pretty standard recipe that I got off the Pioneer Woman, because hey, why not? I used wheat flour, and kneaded the living daylights out of the dough before letting it rise. 
We spent all afternoon occasionally stirring a massive pot of excellent sauce. Just imagine it about two feet to the left, ok? Then there was the grating cheese/chopping peppers/slicing mushrooms/dicing onions part of the day. Look at all the cheese! We had mozzarella, parmesan, romano, and asiago. CHEESE.
This pizza was mine and mine alone: A HA HA HA HA. Look at those excellent crispy edges! Thin crust for the win. 
Here is my preferred method of eating pizza: smother it in arugula. That's one excellent thing about California: there is still available baby arugula in the farmer's market on December 19th. Nice.
