By definition, a food is considered dairy if it is made from a mammal's milk. Milk that comes from cows, sheep or goats is considered dairy, as is any food product that is made out of that milk. In some parts of the world, foods made with camel's milk are common, and those are considered dairy products as well.
Anything made out of a mammal's milk is a dairy product. Butter, cheese, cream, yogurt, ice cream, cottage cheese, sour cream and cream cheese are all common examples. Condensed milk, evaporated milk and powdered milk are all made from milk, typically cow's milk, and are therefore also considered dairy products.
Although eggs are often found in the dairy section of the grocery store, eggs are not considered dairy. They are an animal by-product, but since they're not made from milk, eggs don't fit into this category.
Plant-based milk alternatives are not dairy. Soy milk, almond milk and rice milk are typically safe for consumption by people with allergies to mammal's milk. There are dairy-free versions of butter, cheese, ice cream and other milk-based foods available. They are often made from soy and nuts. Any product labeled as "vegan" is made without dairy products.