Your Dragons... get pets!
Familiars have one purpose - to play with your dragons and give you free stuff every day! This is mostly gold, starting off at 15 treasure per day and maxing out at 50. There's a little box on the bottom right of each dragon's page where you can add a familiar (Change) and bond with them (Heart). The BIG draw is what happens each time they level up - you get a chest! For the first few levels it's Rusty Chests, then Iron, then a Gilded chest when they hit max. These chests can give treasure, gems, familiars, apparel... all kinds of things.
Your Dragons find pets!
There are quite a few ways to get familiars. The main one is from the coliseum. Every creature in there has a corresponding familiar, which has a relatively rare chance of dropping. Once you get a decent level team, however, it's not such a struggle and you can get food, treasure packs and even dragon eggs as a bonus.
Chests are another way - Iron and Gilded chests often give familiars when you open them! Pinkerton will also sometimes give a familiar (I got my first one this way!).
Swipp's Swap Stand offers trades for familiars - some can be got from the coliseum, but some are only otherwise seen in chests. The items for them can be got from gathering and from the coliseum.
The Marketplace has two lots of familiars. The treasure ones can be found in the coliseum or in chests, but the gem ones can ONLY be bought here.
Events - Each Flight Festival has it's own familiar which can only be bought from the Festive Favours tab on the left hand side. There are also a number of Kickstarter familiars, which are very rare and run to 10+ million treasure per familiar. Then there's Boolean, which was only given to beta testers so there's only about 80 of them on the site. Expect to pay a couple hundred thousand gems, or trade for a full set of Kickstarter familiars. There's also been a couple of downtime familiars sent out - Akirbeak, Speedy and Golden Bantam Fangar. They're also limited edition.
Last but not least, Gathering! A number of familiars are available this way, and some of them are extremely rare. Each type of gathering in each element has a familiar, with each familiar usually available in 2-3 areas.