Breed

Once a dragon is fully grown, you have the option of breeding them with a fully grown dragon of the opposite gender. Doing so may produce zero to four eggs. Any eggs produced will usually be of the same breed as either of the parents (the Geode dragon, Two-Finned Bluna and Shallow Water Dragon being the only exceptions, as they can have different parents).

Pairing Dragons
Not all dragons will breed with another species. Two-Headed dragons and Pygmy dragons will only breed among dragons of the same species, and Ochredrakes, Day Glory Drakes and Night Glory Drakes will only breed with each other. Certain dragon species are also incompatible. Golds, Silvers, Magma Dragons, Thunder Dragons and Ice Dragons and all holiday dragons will never be able to breed together, and will always refuse. This is to prevent the chances of always obtaining a rare egg.

Message received and explanation
"The dragons refuse to even go near each other."

This means that the two dragons you tried to breed will never breed with each other. It could be because the two dragons are rare or they just simply refused. It has been found that refusal can happen between common dragon breedings as well so a refusal does not necessarily mean both the dragons are rare.

"The dragons don't really show much interest in each other."

This means that the dragons don't want to breed at the time but are capable of doing so.

"You attempt to breed the two dragons, but no egg is produced."

This means that the dragons bred successfully but no egg was produced.

"The dragons breed and produce (an, two, three, four) egg(s)."

This means they have successfully produced an egg or multiple eggs (up to a maximum of 4).

Geode Dragon
The Geode Dragon can only be made by pairing Stone, Pebble or Geode dragons.

Two-Finned Bluna
The Two-Finned Bluna can only be made from the following:

Skywing x Deep Sea

Skywing x Water

Bluna x Common Dragons

Bluna x Bluna

Goldfish

The Shallow Water Dragon can only be made from the following:

Magi Dragons x Water Dragons

Shallow Water x Common Dragons

Outcome
Once you have found two matching dragons and bred them, they may produce between 0 and 4 eggs. These eggs may be of different breeds, such as half being the same as the father while the other could be like the mother, or all may be the same breed. You may only pick one egg out of the batch; the others will be abandoned automatically and can be picked up by others.

Breeding will have a higher success rate if there are few eggs in the abandoned page.