Plant milk
Milk, according to traditional meaning, is a white liquid produced by female mammals. Since it is an animal product, vegans do not consume any milk or dairy products in any form. Fortunately, milk can be easily replaced by similar liquids made out of plants. There is a wide variety of so-called plant milks: soy milk, rice milk, hemp milk, oat milk, almond milk, etc. Some of them are excellent to use for direct drinking, others suit best for replacing cow milk and cream while cooking.
Production
Plant milk is made by blending grains or seeds (soybeans, rice, almonds. etc.) with water. Extra ingredients might be added to improve the taste, such as sugar, salt, or vanilla. Because of simple production, it is easy to make plant milk at home. Usually one just needs to soak ingredients in fresh water overnight, then cook them, put through a powerful blender, and then filter out hard leftovers. The dry residue can be used for other recipes. To simplify the process even more, one can use soy milk machine (some of them can also make almond milk or rice milk).
Prevalence
Plant milk (especially soy milk) can be easily bought in East Asian and Southeast Asian countries (Japan, South Korea, China, Taiwan, Malaysia, etc.) The reason for this is that it was invented in China centuries ago and spread with its culture (especially Buddhism). Later it became popular in Western countries (especially in North America and Western Europe), and is usually available in most grocery stores as well as in organic&vegetarian shops. It is now popular not only among vegans, but also lactose-intolerant people.
Types
Soy milk
Soy milk is a good and probably most common and cheap replacement for dairy milk. It's quite thick. Most soy milks (at least in Europe) are GMO free, but watch out for genetically modified soy nonetheless, since it is one of the most genetically modified ingredients in the world (the big part of GM soy serves as a feed for livestock though). Beware that some people are allergic to soy milk.
Rice milk
Rice milk is a grain milk made from rice. It's usually not sweetened, the sweet taste comes from a natural enzymatic process.
Almond milk
Almond milk is made out of almonds. Usually it is more expensive than other types of plant milks.
Oat milk
Oat milk is made out of oats and is very common in some countries, for example in Sweden. It is also one of the cheapest types of plant milks.
Related products
Soy milk and oat milk (and perhaps sometimes other types of plant milk) can be used to make yogurt, ice cream, cream and other products.