Some notes about breeding

Use your gems in the first phases of the game to upgrade your hatchery and to get breeding tree and breeding sanctuary.

Do not upgrade the breeding sanctuary too quickly!
The dragons unlocked at level 1 cannot be bred in Sanctuary level 7 and 8. 
As are the dragons unlocked at level 2 in Sanctuary level 8.

Level 1: Deep Red, Wizard, Ninja, Burning (best chances at Sanctuary level 3)
Level 2: Wyvern, Steam, Quake (best chances at Sanctuary level 4)
Level 3: Dujur, Gargoyle, Magnet, Apocalypse (best chances at Sanctuary level 5)
Level 4: Blue, Glacial, Howl, Promethium (best chances at Sanctuary level 4-5)
Level 5: Meteor, Forge, Abyss (best chances at Sanctuary level 5)
Level 6: Supersonic, Sylvan, Dark Elf
Level 7: Mirage, Core, Unity, Acoustic
Level 8: Millenium, Master, Basilisk, Greenfluid

Difficulty groups

Some elements are harder to breed than others. Elements are divided into three difficulty groups:

    Difficulty Group 1: Sea and Nature
    Difficulty Group 2: Terra and Flame
    Difficulty Group 3: Electric, Ice, Metal and Dark

Generations

Generation I: The Elemental Dragons. These Dragons have only one Element.
They can be bred by breeding two elementals of the same element or two hybrids both with the desired element, but they are easier to get by buying them at the store. They cost gold, and not much.
Everyone begins with the basic elements; Terra, Flame, Sea, Nature, Electric, Ice and Metal. Later comes Dark, which is still Generation I dragon.
Light and War elemental dragons (Archangel Dragon and War Dragon) are Generation I-b.
Pure elemental dragons (Pure Dragon) are Generation I-c

Generation II: the hybrid dragons. These dragons have two or more elements and can be bred. Some hybrids are easier to bred than others.
The same way as with Generation I, Generation II-b are the light and war hybrids and Generation II-c are the pure hybrids.

Generation III: the rare hybrids. These dragons have two elements that are opposites. Can be bred.

Generation III-b: Stage 3 rare hybrids. These dragons have 3 elements where two are opposites of each other.

Generation IV: the Legend Dragons. Legend is not an element, so they work as wildcards in breeding. Breeding together two legends results in a random egg. Legends are breedable, you need a Pure/Pure hybrid in order to get a legend.

Generation X or EX: the exclusive dragons

Generation EX-II: The Exclusive Hybrid Dragon. Can be bred. Both parents need to be at level 10 or higher.

Generation EX-II-b: Both parents need to be at level 15 or higher.

Generation EX-II-c: Both parents need to be at level 20 or higher.

Generation EX-II-d: Breedable. Must be unlocked.

Generation EX-III: The Exclusive Rare Hybrid Dragon. These Dragons can be bred using two different Hybrids. Both parents need to be at level 10 or higher.

Generation EX-III-b: Both parents need to be at level 15 or higher.

Generation EX-III-c: Both parents need to be at level 20 or higher.

Generation EX-III-d: Breedable. Must be unlocked.

Generation EX-T: The Exclusive Themed Dragon. This dragon can't be bred.

Generation EX-S: The Exclusive Mobile Dragons. At the moment, there's only the Elements dragon.