Trees are a naturally generated object, they supplies wood. wood is one of the most important resource, you need it to make crafting tables, wooden planks and sticks. To cut them down you don't need any tools but using a axe is quicker. Using a pickaxe works as well but it takes longer and is dipleated quicker. Trees occurs on forest biomes.
Food like apples, oranges, golden apples (rare) and lemons also grow on them. If you mine all the wood away or you mine the leaves with food on them, they will fall down by themselves, ready for you to eat, as well saplings may fall if you place them on the ground, after a while a tree will grow.