“Organism” is a term that may seem intuitively simple, but, in reality, is rather umbiguous. In my humble opinion, an organism should simultaneously display a combination of physiological and morphological continuity (basically, being in one physical piece and having its parts mutually influencing each other), but there is a considerable “grey area”. For example, males of some fish loose all but sexual organs and literally become part of the body of a female, completely fused with the female blood system, etc. Such a hybrid creature may be considered as a single organism (it has one supply of blood, etc.).  In my understanding, unseparated Siamese twins have one organism, but may have individual personalities. It is hard to judge it otherwise, because in some cases one of the twins is represented only by an underdeveloped body part. Sometimes it is difficult to claim whether a virus, a parasite or a symbiont that are present in their hosts are parts of the host organism. In fact, our genome is stuffed with the remnants of viruses and our every cell has symbionts (mitochiondria). A zygote could be just a zygote. It may or may not be an organism by itself. Since we know that the cells within the same organism may have different DNA (see some examples above), than a zygote of placental mammals may be just a cell (albeit, a very special cell) of the mother organism. A free-existing zygote of fish would be a different story. The fact that under some conditions a zygote could develop into a full-size individual animal is not absolutely unique. Sometimes the cells that we call somatic could do it too (naturally in some species, in artificial conditions – in the others). By the way, egg cells of some animals could develop into fully grown individual creatures, so the question of whether eggs are organisms is just as valid as whether the zygote are.   