Hens lay eggs without copulation. These eggs are sterile: they cannot develop into an embryo and thus another hen. So, to some vegetarians, they are acceptable. They are non-viable.

But hens reared in the presence of a cockerel may well be fertilised, and thus viable, and thus not acceptable for very strict ovo-lacto-vegetarians.

Ducks, quails, and some other farmed birds, do not generally lay eggs unless they have copulated, and so their eggs are often not considered suitable for vegetarians.

Birds' eggs can be collected without harm or suffering to the animal; so, some vegetarians considera that fine.

But eggs of any type are quite definitely an animal product and so not edible by vegans, whether viable or not, fertilised or not.

Fish eggs are never suitable for vegetarians, because collecting them means killing the animal. This is a simple and obvious distinction but one that is not apparent to most non-vegetarians. So, no caviar, no fish roe, etc.

For the very strict, that also means not consuming a lot of beer, because it is fined with isinglass, a byproduct of killing fish. However, I know vegans who drink beer.

It is a lot more complicated than it appears to outsiders.