Coins are earned as you progress through the game and once you have accumulated enough of them they can be used to unlock new dragons or equip new costumes or pets.
Gold Creatures
You can earn a decent amount of coins by eating any gold-coloured creature you encounter, this also includes golden bugs and golden people. You will not have any problem finding these gold-coloured creatures as their shining gold makes them easy to spot. However, if you do find one and notice that it is precariously placed and won't be easy to get to because you need to get past mines or some other harmful obstacles you are better off playing it safe and leaving them alone rather than taking the risk of getting blown up by a mine for instance.
Missions
The fastest way to earn plenty of coins is by completing any of the missions which you can view just before the start of a run. These missions will normally involve a task such as scoring a certain number of points or using your Fire Rush a certain number of times. Although some of the missions may appear difficult to complete you need to remember that they do not have to be accomplished in one run as they are cumulative.