Let's talk about Designer Breeds, Breeds, and Varieties, so you'll understand the differences in terminology.
There are two main differences between Designer Breeds and real Breeds: Designer breeds will not breed true, and they are not American Poultry Association (APA) recognized.
A Designer Breed of chicken such as a Favaucana is not an APA-recognized chicken breed, the same way a Goldendoodle---crossing a Golden Retriever and a Poodle---is not an American Kennel Club (AKC) recognized dog breed. That is to say: they are recognized in the sense that you've probably heard of them, but Favaucanas and Goldendoodles are both simply crosses of two other recognized breeds.
Designer Breeds do not breed true. That means if you breed Goldendoodle to Goldendoodle, the puppies will vary: some will have very curly fur, some will have wavy. Some will have no shedding, while some will still have low shedding. Likewise, if you keep breeding Favaucanas together, you'll get some with single combs laying brown eggs, like regular Faverolles. Some will have pea combs and lay blue or green eggs. Some will have four toes, some five, etc etc.
Real breeds must breed true: their offspring must have the same qualities as the parents. That can be done in some cases, but it takes time and effort.
Making a "Real" Breed
If we wanted to make Lissa's Favaucanas into a real APA breed, for example, we'd first have to breed them to a point where they breed true. So, we'd have to select pea combs for a few generations and breed out the recessive genes that would lead to a single comb... and we'd have to keep an eye to make sure egg color remained correct, that leg feathering and number of toes was consistent. To get them approved by the APA, we'd then have to show there was an interest in having Favaucanas; that multiple people were raising and showing them. We'd have to submit standards and have those standards approved.
Lissa's Wheaten Favaucanas
For now, Favaucanas, Carolina Blues and so on: these can be called crosses, custom crosses, varieties, or designer breeds. They are not actual breeds, per se.
Now, there's nothing wrong with a Designer Breed or cross-breed---Red Stars are an old, popular cross originally made for production purposes, for example. They're great layers, and as it turns out, they're super friendly, so have even become popular with backyard enthusiasts, too!
The main issue is really just that cross breeds sometimes can't be exhibited at shows, unless there the show has a category for unrecognized varieties. Don't show birds? Then it really doesn't matter if your bird is recognized or not. The other thing to remember is that if you intend to hatch your own fertile eggs, you may not know exactly how the babies will turn out. That can be deeply fun, though! It just depends on what you want for your flock; it's a personal decision.
A regular breed of chicken is one that breeds true and is recognized (in this country) by the American Poultry Association (APA).
Complicating the term "breed" is that there are real breeds that are recognized by poultry orgs in Europe, but not in the US (yet). For example, Swedish Flowers are not yet recognized by the APA, but they do breed true and are considered a breed in some other countries. It's okay to call Swedish Flowers and other European breeds "Breeds." But be aware that if you want to show unrecognized breeds, you'll have the same issue as you have with Designer Breeds: you can only show them in a category that allows for the fact that your bird is a breed that is unrecognized here.
Further, even when a breed is recognized, sometimes there are plumages available in Europe which are not official here (yet), such as Jubilee Orpingtons, Red Sussex, etc etc. Sometimes new colors that have just been developed by enthusiasts, and are not recognized anywhere, yet. This presents the same problem: if you want to show these rare colors, you'll have to do so in categories that allow for that.
Chocolate Eggers, including Rare Marans varieties
Just a few years ago, for example, Silver Cuckoo Marans were a recognized APA color, but Black Copper Marans were not! Then the APA approved Black Coppers, hooray! But Golden Cuckoos, Blues, Splashes, Blue Coppers and others are still not officially recognized Marans colors at this time. It's safe to say that more official colors are likely to be approved in the future, given the interest.
The word "variety" can be used to describe designer breeds, but it can also be used to describe different colors of a real breed. So, in your flock, you may have three different varieties of the Wyandotte breed: Columbian, Silver Laced, and Golden Laced.
Here is a list of official APA breeds and varieties. The latest version the APA has listed online is currently their 2012 list.