Super Mario Run contains a whole host of enemies from the Super Mario World.
If Mario touches an enemy, he will lose a life. Short enemies such as Goombas can be vaulted.
Mario must either dodge enemies when running through each stage, or kill them by jumping on them. If Mario is invincible (on a Coin Rush), he will be able to run straight through any enemy and instantly kill them.
When Mario kills an enemy, you will gain coins. At the start, every enemy will give you 1 coin. Killing enemies will increase the guage next to a picture of the enemy at the end of the stage. When this fills up, that enemy will level up and you will receive an extra coin per level when killing that enemy.
The maximum level is level 3, where you will receive an extra 3 coins (for a total of 4 coins) for killing that enemy.