Strong against: Low HP, low armor, low power armies.
Weak against; Things that stun(e.g. rogue), dodge(e.g., rogue), or 'incapacitate'(e.g. geomancer),
Best Paired with: Healer, abbey, and artificer all work decently with the dragon.
The dragon is one of the few units I recommend playing before deployment cost reducers like the general are down, if you have to make such a choice. This is because the dragon becomes increasingly feeble as more units enter the field; after a certain point, your dragon will probably be annihilated right after sticking is smoke-exhaling nose near your opponent's side. It simply will not be able to take down units fast enough.
Its effective attack range of 6(including 3) and movement speed of 3, however, makes it excellent for putting very quick pressure on your opponent; low power, low HP units are typically the first units your opponent will deploy. With some luck, you'll be able to land a crushing explosion on a white or black mage, along with a number of fodder units.
Unfortunately, even with only 2-3 turns before your dragon is a threat, your opponent will typically be ready for it.