Here's a hypothetical example.
Let's say you have a bunch of white chickens, and you hatch some eggs from them, and one of those hatched chickens is bright blue. That's pretty cool, you want more blue chickens. That blue chicken grows up into a rooster. You mate him with some hens, but all the eggs hatch into white chicks, because the blue gene is recessive. You don't know which hens have the blue gene, except that you know that the hen that laid this blue-chicken egg must have the blue gene. Therefore, you mate him with his mother, and some of the eggs that result from that hatch into blue chicks. You then take those blue chicks, which are siblings, and mate them with each other, until you have enough blue chicks to stop breeding closely related ones. If you only have a couple generations of inbreeding, it's generally harmless enough, and now you have blue chickens.
Stepping back farther into history, suppose someone came back from a distant land with an exotic chicken breed. Only a few made the voyage. Then you only have a few to start with. Entire breeds have started this way.
In other cases, there are only a handful of individuals to work with, at a breeders disposal, that would be considered pure. Only a few of those have the desired traits. The best way to fix traits, as illustrated above, is to "line breed" (the term we use when inbreeding works out well) . Inbreeding fixes traits, that means it sets them, as in affixed, it doesn't fix them as in improving them. If bad traits are there, as they were in some of the royal families that gave rise to the notion that inbreeding was a bad thing, the bad traits will be magnified just as the good traits would be.
There are many families of chickens that are highly inbred, and nothing is broken, so nothing needs to be added. Doesn't take much to revitalize a line if it gets too close, and often takes further inbreeding to weed out any bad traits that were introduced with the fresh genetic material.
I personally have chickens that are highly inbred, and there are hens within my line that are hatching chicks from their own eggs as teenagers. They have been hurt in no way, because I and others have been careful not to breed anything showing the slightest hint of bad traits. The bad traits simply aren't there, and with no new genetics, there is nowhere for them to come from besides a mutation, which would be weeded out if it occurred.
People have the notion that you can't breed a rooster to his kin, when in reality, there are commercial production chickens that have been carefully linebred for close to a hundred years with no outside introductions.
The term used when breeding chickens is “line breeding”
It’s is a common practice among hobbyists who breed their birds for showing.
When like breeding you can focus on certain points to improve.
Eventually you will have the birds near perfect and you will be able to trace them back for generations.
People may also line breed to stop accidentally breeding in a fault to their own line.
Brother to sister is not ok, as they are too closely related.
Mother- son etc is ok