Before we get too far into the specific dashis, I want to back up a bit. I have a question for you. When you think of the basic tastes, what comes to mind? If you are from a western country, you will most likely have answered sweet, salty, bitter, sour, and perhaps spicy. If you ask a Japanese person, they will tell you that there are 5 tastes: sweet, salty, bitter, sour, and umami.
Umami is the taste of nucleotides or glutamates hitting your tongue. The body needs these building blocks of protein to survive, so we are hard wired to enjoy this taste. But what exactly does umami taste like? The closest word we have in English is probably savory, but that doen't quite do it justice. Umami rich foods include meat, seafood, including combu and other seaweed, and vegetables like ripe tomatoes, shiitake mushrooms, Chinese cabbage, as well as other items like parmesan cheese and soy sauce. When mutiple umami rich foods are combined, some kind of magical umami multiplication effect occurs and the total taste becomes far greater than the sum of the ingredients. Because of this, a mix of fish and combu dashis forms the flavor base of many Japanese dishes.
Now to the specific dashis. Katsuodashi (かつおだし, カツオだし, 鰹 だし) made from dried bonito fish, and is the most widely used. Similar, but less widely used is nibodashi (にぼしだし, ニボシダシ, 煮干出し), which is made from dried anchovies. If you are vegetarian or vegan, you should make friends with kombudashi (昆布だし), made from dried kelp, and shiitakedashi (椎茸だし), made from dried shiitake.
Dashi is considered an essential base to all kinds of Japanese dishes, including pretty much every kind of soup (miso, soba, udon... ), nimono and more less anything cooked in a broth. As katsuodashi, or some combonation including katsuodashi is generally used, be very careful when eating in a restaurant or buying prepared food from a combini or grocery store. When you cook at home, you can use combudashi, shiitakedashi or a both to create a rich base flavor for your dishes. When buying dashi in a grocery store you have two basic options. You can buy actual dried combu or shiitake, and make the dashi yourself, or buy powdered dashi, which includes individually wrapped portions in little paper packs. These days, most Japanese people buy the powdered version, as they require no work, but the traditional version is healthier (the powdered version also contains salt, sugar, and other additives) and unambiguously vegetarian and vegan friendly. If you are buying the powdered version, check to make sure that the your combu or shiitake dashi doesn't also contain katsuo. Also, the instant version will usually contain amino (アミの酸), amino acids from which source (animal vegetable or mineral?) they don't bother to write.
This brand is vegetarian, other than containing the ambiguous amino.
How to prepare combudashi and shiitakedashi
Using dried combu or shiitake to make your own dashi is really quite easy. The standard way to make combudashi is to wipe a piece of combu with a damp cloth, then soak in a pot of water for at least 20 minutes. Turn on the heat on low and remove the combu from the pot just when the water comes to a boil. You can save the combu you used for your stock to use in other dishes.
When I make miso soup, I just throw a piece of combu in before the vegetables, not bothering to presoak cause I'm a lazy cook. You can remove the combu before serving, but it's quite nutritious, so I like to eat it along with my soup.
This combu is cut into convenient dashi making strips.
For shiitakedashi, the recipes I've seen really vary. Start by rinsing a handful of dried shiitake. You can get a fair amount of flavor by simply soaking the mushrooms in water (for a half hour, a few hours or overnight), or for a stronger flavor, simmer the shiitake for about 20 minutes. You can then use those mushrooms in other dishes. Shiitake dashi is often made with both shiitake and combu for a richer flavor.