A good many of our fish have been bred back and forth to obtain brighter colors, longer fins. I have been doing this with my bristlenose to obtain longfins, but they are still the same species of fish. Some fish are going extinct in the wild, like Endlers, and can breed with other fish,like guppys, and if allowed to breed indiscriminately will disappear from the earth. That's why I encourage everyone to keep their guppys and Endlers separate even though they are in the same family. I'm just really conflicted about fish that are true hybrids (two different fish) and I wouldn't take it upon myself to tell someone they were wrong to own them but I just am not sure we have the right to create new fish. Just my two cents worth.
I don't think there is a problem with having hybrid fish. I don't like the massive amount of breeding/culling that's done, but the fact that fish breed in such large quantities makes it less objectionable to me than when they do the same thing with dogs/cats.
Sometimes its accidental, as timmy-man said. I've heard that it can happen really easily in livebearer tanks.
It happens really easy in Africans cichlids from Lake Malawi...I actually had a Hap brownae from Lake Victoria hold for a Red Top Zebra from LM (babies didn't make it)...a mouth brooder can very easily be holding for the wrong type of cichlid. Especially if you don't have a female for your particular species.
The reason there is 1000s of species of cichlids in LM is because of the cross breeding...but that happens in their natural habitat so it is only hybrid if man is involved somehow.
I have and had several hybrids.
I have a young batch of hybrid marble yellow mollie cross
I had a zebra danio crossed with a long fin zebra. Its tail was half short and half long but it swam just fine.
Cross breeding animals of the same species (such as dogs and wolf for huskies) or breeding the same species to get a specific result (such as chihuahuas to keep getting them smaller and smaller) is NOT creating hybrids. A true hybrid is when you get two DIFFERENT species to procreate a NEW species (for example horses and donkies to get mules). However, the one side effect that holds true for ALL true hybrids is that they are not able to have offspring of their own. This holds true no matter what species of animal or plant it is. So its not like if you make hybrids they will one day take over the world. Cross breeding, however, can make new looks of same old species or more stronger ones (for example, some cross polination made new kinds of wheat long ago that make more seeds, but, they are not as healthy). Usually, however, crossbreeds are not as healthy as its predecesor, thought not always.
(For you scientists out there, I'M NOT ONE.. duh! However, you can testify to the veracity of these statements regardless of my using the correct scientific terms or not)
bhcaaron has a good point, although I just discovered, through a bit of research, that he's only partially correct.
First, let's define somethings. Species is the most specific name for something (look at the words "species" and "specific"). For example, lyretail molly is a species and a balloon molly is a species (albeit an artificial one). Species is sometimes determined by two similar animals that are capable of cross-breeding easily, but, for some reason, don't do so in the wild. Genus is probably the word that bhcaaron was looking for. Genus is the name of a group that encompasses a number of species. For example, mollies.
There are different levels of hybrids, such as interspecific hybrids (which would be the dalmation molly, a cross between black and silver or white mollies, if I understand correctly), inter-genus hybrids (this would be the liger or mule, I believe). Every once in awhile, there is an inter-family hybrid, which are really rare.
So this means that a cross between a short-finned danio and a long finned zebra danio technically is a hybrid, just not a very odd one. A cross between a molly and a guppy, if I understand it, would be an inter-genus hybrid, which, while relatively easy (kind of like a mule), is a bigger deal.