Mommy...?
Find your mother


Baby's First Minion
Make a friend


Growth Spurt
Grow larger


Adoption
Retrieve the green dragon egg


Goblin Killer
Defeat the goblin chief


A New Life
Hatch the egg


Monster's End
Destroy the black knight

If you do nothing particular, you will lose your green friend at the end, thus giving you this achievement after ending the black knight's existence.

Lonely Victory
End the game with no other dragons in your group


As the game saves just after acquiring this achievement, you have to do a second playthrough. By triggering a special event, you have access to 3 linked achievements. Here's is how to proceed:

When the Black Knight threats the lizard group in the mushroom cave, your red scarf hero wants to defend them. This time you don't want him to die. Keep the knight busy while fighting against you until the drawbridge is risen completely. The heroic lizard will try to escape. Just stand next to him and serve as a mount.

You may need the "help" of the knight, cause the dude is really heavy!
After the rescue, you will be granted this achievement:

Protector
Save a friend


Later, in the second dramatic scene, you normally will leave a partner threatened by a skeleton warrior. However this time, in this second playthrough, the red scarf hero will help you. While switching on the "vacuum device", you have to quickly breathe fire into it. Saving the little guy will come with a cost, since you will loose your ability to breathe fire from now on.

Sacrifice Of Flame
Give up your ability to breathe fire


Finally, when it comes to the end, you will have the chance to save the green dragon (yeah!). In fact, your loyal red scarf dude will luckily open the gate in time, thus saving the poor creature. In order to the green dragon to avoid death, you may have to spare the three goblins life in the previous room. Finally, you must enter the room as soon as possible, to make the little dragon fleeing. (Thanks Taran, SyperTechBone and Okuraorca for the tips.)
Beat the final boss again, but this time with no ability to breathe fire. Stomp and bite him to the death! Good luck!

Hero
Reach the surface with the green dragon

Ten Second Rule
Eat food off the ground


Glutton
Eat food while at full health


Swift Strike
Bite a ghost skeleton out of the air


Eating On The Fly
Eat a creature before it hits the ground


Overkill
Kill a mouse with a direct hit from a fireball


These achievements are self-explanatory and easy to get. The last two achievements may require a little of practice. You may burn the creature and trying to eat it on the fly (Burn>Jump>Bite). For Overkill you need a direct hit, spash damage apparenly doesn't count. Hitting while being in air may be easier to land a direct it. (Jump>down+fireball).

Kill It With Fire
Kill 50 enemies with fire


You Are Crunchy And Taste Good With Ketchup
Eat 20 Goblins


Stronger than Stone
Smash 10 pieces of masonry


Laid To Rest
Kill 50 undead enemies


I might call them "grind" achievements but they are actually easy to do and they come naturally while advancing in the game and making sure to defeat any enemies you cross.

Stoned
Kill and enemy by dropping a stone on them


This one is actually easy if you pay attention to a certain place where a skeleton awake and stand under a breakable floor. (screenshot below).


Dragon Rider
Give the kid a ride


This has to be done in the very beginning in the village. A little boy will approach you. Just turn your back and sit down (press down during a long time).

Air Born
Stay in the air for 10 seconds


It's recommended to try this after getting the last air upgrade (see the achievement below). Find a corridor (screenshot below) and clean it from enemies. Each time you jump, press the button down for several seconds for the dragon to glide and when the glide ends jump again. Move from right to left and left to right, then right to left again.


Maximum Power
Increase all abilities to maximum


You can't get it along with "Sacrifice of Flame" so try it in the first playthrough. (thanks fouyo) Here are all of the locations.

First Cave: 1 Air, 1 Fire, 1 Health

Mushroom cave: 1 Fire, 3 Health

Goblin Cave: 1 Health, 1 Air

Skeleton Cave: 2 Health, 1 Fire, 2 Air