There's 5 elements/colours. Each element has 5 different runemals. For this quest you need to capture all Fire(red) runemals. You can either trade 15 diamonds and hope you get a runemal you don't have or you can try to find a runemal (after a battle) and capture him with one of the two runes (35% chance of capture and costs 5000 coins or 85% chance of capture and costs 10000 coins). Runemals appear at any location, but chances are highest if you try to find it in it's corresponding area. I Highly suggest using the 85% runemals, as I've only used these to capture runemals. Never managed to capture one using the 35% one.
The list of runemals is:
-(don't know the name but it's a rat)
-(don't know the name but it's like a salamander with a large tail)
Capturing all 5 of one elements rewards you with one achievement (so a total of 5 achievements.