There are three basic Dragon types-Fire, Water, Earth (AKA Wood). There are specific advantages and disadvantages to each type. These are only the basics. There are many more types, but those come around later.
Fire:
The Fire Dragon comes from a red egg.
Advantages: Fighting Earth
Disadvantages: Fighting Water
Water:
The Water Dragon comes from a blue egg.
Advantages: Fighting Fire
Disadvantages: Fighting Earth
Earth:
The Earth Dragon comes from a green egg.
Advantages: Fighting Water
Disadvanages: Fighting Fire