There is no officially sanctioned breeding of dragons, as dragon breeding was outlawed by the Warlocks Convention of 1709.
However, they have been known to interbreed, producing rare hybrids.
Below is a list of the ten known pure-bred breeds of dragons according
to Fantastic Beasts and Where to find the,:

• Antipodean Opaleye
• Chinese Fireball (Known as "Lion Dragon")
  
• Common Welsh Green (A native dragon of Great Britain)
  
• Hebridean Black (The other native dragon of Great Britain)
  
• Hungarian Horntail (Considered to be the most dangerous dragon ever)
  
• Norwedgian Ridgeback
• Peruivian Vipertooth(A highly venomous dragon)
  
• Romainian Longhorn
• Swedish Short-Snout
• Ukrainian Ironbelly (The largest dragon species ever recorded)
  

There are two other breeds, according to Dragon Breeding for Pleasure and Profit.
It is unknown if these are pure-bred dragons, hybrids or, as the book
is out-dated, these dragons may have become extinct due to illegal
dragon huntings.

These are the two dragons:

• Catalonian Fireball
• Portuguese Long-Snout