Vegetarian diet include diary products and eggs, however vegans don't eat any animal products - beef, chicken, fish, cheese, milk, etc. Some vegans don't eat honey, and some won't wear/use leather, silk, or down.

People have different reasons for being vegan; usually for the environment, animal rights (www.veganism.com/ ), or health (or some combination.) There are many environmental reasons to be a vegan. Instead of using land to grow food to feed cows (which can be eaten by a small number of humans or produce dairy which humans eat), this land can be used to grow food for humans directly, and a lot more people can be fed off the same amount of land. The production of meat requires large amounts of water and energy, and pollutes water, and possibly air. Vegans are making the choice not to support any of these industries. Meat and dairy products are often produced in particularly unhealthy ways. Animals may be fed antibiotics or other animals, and this can affect the health of people who eat these animals or the dairy/eggs they produce.

More information is available at:

www.goveg.com/feat/a-favs-intro.asp  (an introduction to veganism, which explains different reasons for veganism)
veganoutreach.org/whyvegan/  (see sections on Environmental Destruction and Resources & Contamination)
www.vegan.org/about_veganism/  (has sections on animals rights, and environmental and health reasons for veganism)

There are some things to keep in mind when cooking vegan on biketour. So long as you prepare a variety of vegetables, getting the necessary vitamins shouldn't be a problem. It's important to get enough protein, and to eat substantial food so that people are satiated after a long day of biking.

B12 is one food which doesn't naturally occur in plant-based foods. Many foods, such as nutritional yeast and many brands of soy milk are fortified with it. From what I understand, our bodies can store B12 for years, so it isn't really a huge problem if people don't get a lot during the ~6 weeks of biketour. However, B12 is something vegans should consider. All vegetarians should also be aware of essential fatty acids, which non-vegetarians can get from fish. Many nuts/seeds, and their oils (such as flax/linseeds and walnuts) contain these fatty acids, so it's not a big concern so long as people eat a diverse diet.

For breakfast, there are a number of different spreads for (preferably non-white) bread, such as nut/seed butters (peanut butter, tahini, etc...), jams, margarine (margarines containing hydrogenated oils should be avoided), nutritional yeast, and mustard. Fruit or veggies will also round out the breakfast. Diversity is important, and it's nice to eat different foods. It's easy to make oatmeal with oats and soy milk (powdered or not), and this is even better if mixed with nuts, seeds, and dried fruits.

Homemade bread spreads are great too. Hummus is made by mashing chickpeas, garlic, tahini, lemon juice and olive oil. Other bread spreads can be made by mashing up beans, lentils or tofu with some spices and veggies.

For dinner, the meal should be pretty filling. It's good to have a grain, lots of veggies, and a protein source.

TVP (textured vegetable protein) is a dehydrated soy product, available in little pieces and bigger chunks. It can be added to a stew, re-hydrated and then stir-fried, or cooked in other ways.
Beans are good sources of protein. Dry ones take a very long time to cook, and buying pre-cooked canned ones is wasteful (in terms of packaging and cost). Dried ones can be soaked on a rest day, or on the trailer or individual water bottles while biking, and then they don't take so long to cook. Some, like mung beans, take less time than others, like chickpeas/garbanzo beans.

Lentils are reasonably quick to cook, particularly red ones. They should be put in a stew before most veggies, but they don't need to be soaked or anything.

Tofu or other soy products are good sources of protein, and filling.

Tempeh is an Indonesian fermented soy product (although it can be made with other beans.) It is really healthy and tasty, but I'm not sure about the availability, price, and packaging.

Seitan is a high-protein meat-like product made from wheat gluten. Again, it might only be available in small, expensive packages, although some innovative folks could make it from scratch on a rest day. 

Nuts and seeds have lots of protein, and can be added to dinners. I like to cook with nut butters. It is a bit easier to distribute them in a stirfry if they are mixed with hot water first, but that isn't necessary.

Coconut milk can also be added to stirfries to make them more filling.

Eat lots of different vegetables! Different veggies require different cooking times. Potatoes take a while, while thin greens like spinach take almost no time. Except for potatoes, eggplant/aubergine & really hearty greens, most veggies can be eaten raw, so it isn't really a problem if you don't cook them exactly right.

Rice is a staple grain which can be bought easily; brown rice is healthier. However, rice can be a bit difficult to cook, and variety is good. Some other grains are less sensitive and turn out okay if just thrown into a stew. Quinoa and teff are high in protein, and easy to cook.
Kasha/buckwheat is good, as is polenta (make from corn). Couscous (which is actually pasta) and bulghar wheat are both super quick, and you can cook them just by putting boiling water on top and covering it.

Spices will add taste. Here are some examples: pepper, paprika, soy sauce, curry, ginger, garlic, nutritional yeast, basil, oregano, cumin, tumeric.

Nutritional yeast comes in yellow flakes and is sort of cheese-like; it is really healthy and really tasty. I'm not sure how widely available it is---German health food stores carry it, but I'm not sure about Eastern Europe.

To make a big stew, you'll probably want to fry in oil (which will also add some necessary fat). Peanut oil and canola/rapeseed are both good for cooking at high temperatures. Olive oil tastes good, but it's not ideal for cooking at really high temperatures.

Another food tip is that cornstarch thickens to make sauce or thicher stews (as do arrowroot powder and kudzu root, but they are harder to find.)

Add oil to the pot, then onions, garlic and ginger. Add a bit of water and some lentils and grain (such as quinoa); cook some more. Add various spices, such as cumin, red pepper flakes, mustard seeds, salt, soy sauce, and basil. Add root veggies (like beets, carrots). Add thicker greens, like kale, and then brocoli, cauliflower, and/or summer squash; cook some more, adding water if it looks dry. Mix up some peanut butter and hot water, and add that, with more spices and maybe some vinegar or lemon juice. Add some tomatoes, and spice to taste.

There are plenty of treats for vegans, although probably not for biketour to buy. Halvah is a nice sweet, which is sometimes made with egg white and sometimes not. Dark chocolate is usually vegan.

It's better (on biketour and in general) to eat food which isn't processed and wrapped up with a long list of chemical-sounding ingredients. However, if one is looking at a label with list of ingredients, there are some animal-based ingredients to watch for. These include anything with gelatin, whey, casein or bonito (a fish product found in some asian pastes), as well as more obvious things like eggs, dairy (for vegans), and chicken broth.