Rane, I'll try to put my understanding of this into words because I feel that there is deep misunderstanding around this topic in some circles. That being said, I only have high-school biology to draw upon, so feel free to correct my own misunderstandings or murky reasoning!
Genetically modified organisms are plants, animals, bacteria or virus that have been altered (by humans) at the microscopic level of DNA. Deoxyribonucleic acid, that old double helix, is a natural information storage system that encodes the form and development of all organisms (excepting most virus). This system of information storage is cool because not only is it physically shaped to be very resistant to damage, but it's double structure provides a back-up in the case that damage does occur. In the world of computer language we use a binary system to express all functions, that is, everything is a 1(on) or a 0(off) and by linking together incredibly long chains of these digits we can do stuff like talk to each other on the internet. DNA goes one step further by using a quaternary system where everything is expressed as 1,2,3 or 4. By using incredibly long molecules of these "digits" (actually different types of sugars) DNA can be made into awesome things like sea anemones, chicken-hawks, lactobacillus bacteria, or you and I.
Now, out in nature, only DNA that speaks roughly the same language will be able to combine successfully to make these creatures. We call these 'language' divisions species. A horse and a horse speak the same genetic language and therefore their DNA can combine to create another horse. One horse's 'accent' may differ (a shetland pony vs. a clydesdale) but both of their genetic structure is still written in horse, no matter what the individual traits of each horse is. The result of a shetland-clydsdale hybrid would be an odd recombination, but no different from a light-skinned dark haired human swapping genetic info with a red-headed albino. Sex between same specied organisms usually works. The same goes for breeds of; tomatoes, apples, dogs, or starfish. Point is: a starfish cannot breed with a tomato to create a starmato-fish... until we came along.
There are cases where two organisms of slightly differing species can intermix. Horses and donkeys are different species, but they are from a common ancestor species that split apart millennium ago. Therefore they are similar enough to produce a hybrid species; the mule. It is a great note that mules are notoriously sterile. That is, they are DNA mute and cannot pass genetic information to each other. Bacteria are another strange case, they do not recombine DNA like us but mostly clone their identical structure over and over again. Not only do they count on random mutation (ie: damage to their DNA that by happy coincidence, still works well or better as the original copy) but recent study has found that they can actually swap select bits of their DNA (we call these discreet bits 'genes') with each other. What they are doing there is more like what we do with GMOs than most people would like to admit.
What we are doing with GMOs is creating that fabled starmato-fish. We have learned enough about genetic language to be able to read the parts of DNA and understand what they are supposed to do. We take a small snippet of DNA from say, a deep-sea flounder and paste it together with a strand of soya DNA. We then get a Soya plant that has the ability to freeze solid and not die. We take a snippet of fire-fly and paste it into a pig. We then have phosfluorescent pigs. Seriously, we have done this. Mostly, the reason we have done this sort of thing in our food crops is to be able to get more production out of our plants and animals. Being able to ship without damage, being able to pick before ripe, being able to withstand weather. What we have not done is care what the resultant effects of this gene-swapping is on the rest of the subject species. Will that starmato-fish interbreed with regular starfish and permanently change the species? Will GMO corn pollinate regular corn and change it too? Other questions are: How does it taste, how nutritious is it, how does it effect the persons or animals that eat it?
Some say that, like a tomato planted in the forest, GMOs have no chance of survival in the real-world. Natural species have evolved to fit specific ecological niches and have proven their ability to out-compete their natural rivals. We humans create these little artificial ecologies called gardens or farms were all the freaky mutant plants get to survive. Mostly we have created these mutants (non-GMO hybrids) to get better flavour, larger fruit, as well as better pest, weather and drought resistance, but we have done it in ways that could conceivably happen in nature. So far I have seen little interest from GMO manufacturers in bettering flavour, nutrition (golden rice being one exception), or bio-diversity. They are generally more interested in profit margins, mono-cultures and copyrights.
Until I see that mind-set change I will do my best to not consume GMOs nor feed them to my animals.