Breed

Once a dragon is fully grown, you have the option of breeding them with a fully grown dragon of the opposite gender. Originally, doing so would produce from zero to four eggs; at present, to reduce the number of abandoned eggs, this has been changed, and breeding produces at most one egg. 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, in the case of Pygmies, Pumpkin Dragons). Ochredrakes, Day Glory Drakes and Night Glory Drakes will only breed with each other. Previously, RarexRare breedings were always refusals, but as of June 10th, 2011, they are able to breed together, with the chance of producing a rare egg the same as the chance when breeding with a RarexCommon pair.

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 can happen between dragons of any rarity. Breeding Holiday x Holiday or Retired x Holiday any time besides at least one of the Holiday's breeding season will result in a refusal. (Ex: Christmas x Halloween during the Valentine's breeding season, Frill x Valentine at anytime besides the Valentine's breeding season)

As of June 10th, 2011, dragons that receive this message will be deleted from the breeding list.

"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 egg."

This means they have successfully produced an egg.

Geode Dragon
The Geode Dragon can only be made by pairing Stone, Green 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

Shallow Water Dragons
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 only one egg. This egg could be the breed of the father or the mother. If either parent is a holiday dragon, the egg will only end up as such on said parent's holiday. Otherwise, it will be the breed of the other parent.

The chances of a successful breeding are not affected by how many eggs are in the abandoned page, which is a common myth.