Plant milk is a milk-like liquid made out of plants (seeds, grains, nuts, legumes). There is a wide variety of plant milks (or milk substitutes): 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.
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).
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.
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 is a grain milk made from rice. It's usually not sweetened, the sweet taste comes from a natural enzymatic process.
Almond milk is made out of almonds. In Europe it is usually more expensive than other types of plant milks, while in US prices are generally lower, making it quite a popular choice.
Hemp milk is a drink made from hemp seeds. Hemp seeds contain a three-to-one ratio of omega-6 and omega-3 essential fatty acids and other nutrients include magnesium, phytosterols, ascorbic acid, beta-carotene, calcium, fiber, iron, potassium, phosphorus, riboflavin, niacin and thiamin, all of which make this plant milk a valuable product for vegans. Hemp milk also contains 10 essential amino acids, making it a good vegan source of protein as hemp protein does not contain phytates, enzyme inhibitors found in some soy protein that can interfere with the absorption of essential minerals. Hemp protein may also be more digestible than soy protein because unlike soy, it doesn't contain oligosaccharides, complex sugars that can cause flatulence if not properly broken down during digestion.
Hemp milk is generally more expensive, but can be made at home without greater difficulties.