Woodcutting is the skill that is based on chopping down trees with an axe. Xp is gained by chopping down trees using an axe. You can also gain xp by destroying any giant mushroom with an axe as well.
Perks of leveling up woodcutting:
Tree Feller: Cuts down an entire tree whenever chopped with an axe. The ability can be activated by right clicking with an axe in hand and hitting a tree. This ability lasts for 2 seconds base and increases by 1 second every 50 levels.
Leaf Blower: Unlocked at 100 woodcutting removes all levels instantly from a tree.
Double drops: Grants a chance to get double drops from chopping logs or giant mushroom blocks. The chance increases by .1% per level of woodcutting up to a cap of 100% chance at 1000 woodcutting levels.