I own and operate a certified organic farm. I am legally required to make sure my crops do not get contaminated by GMO crops. I have a personal preference not to consume GMO crops. So there's my background and my perspective.
The most common GMO crops are corn and soybeans. GMO wheat has been created but is not currently marketed. There are GMO potatoes now too, so they may become more readily available. If you are eating conventional food made from corn or soy, in the USA you are almost certainly eating GMOs.
GMO corn and GMO soy are genetically modified to insert genes into the plants to resist a weed killer called glyphosate. Conventional farmers like this trait because it allows them to plant the crop then come back later and spray the entire crop with a weed killer. Then most of the weeds die but the corn or soy doesn't die. This saves money in growing the crop, because the farmer doesn't have to till the weeds to kill them.
In corn they have also inserted the genes from a bacteria, so that when when pest insects try to eat the corn, they die from the bacterial toxin present throughout the genetically modified plant.
All this insertion of genes happens quite differently from the traditional methods of plant breeding that has been going on for millennia.
So when you are assessing the safety of consuming GMO crops, you have to look at the entire picture of not just the crops, but the chemicals used on the crops. There's no way to completely eliminate all the agricultural chemical residues from the food. If you eat conventional food, you are exposing yourself to the chemicals of industrial agriculture, in trace amounts.
Glysophate is probably the main chemical of concern. It has been shown to cause tumors in laboratory animals. It is classified as a probable carcinogen. If you eat conventional corn or soy, you are ingesting trace amounts of a probable carcinogen. Glyphosate is the active ingredient in Round-up weed killer, so you might be buying this probable carcinogen to spray around the house, too.
Glyphosate has been around long enough that weeds have developed resistance to it. So now Dow Chemical has used GMO techniques to insert genes into corn to make it resistant to 2-4-D. You may remember this chemical name because of Agent Orange, a Dow Chemical herbicide that was widely applied by US troops in the Vietnam war to defoliate the jungle so they could see where to bomb the Vietnamese people. The troops who worked with Agent Orange got cancer from their exposure to it.
So if you eat GMO food, just be aware that you probably are eating food that has bacterial toxins inserted into the DNA of the plant, and the plant also has genes within it to resist industrial chemicals such as organophosphates and neonicotinoids, and that the food likely has trace residue amounts of these chemicals in it. The question of the safety of humans eating this stuff may take science decades to determine.
My personal preference is to avoid all that as much as possible. Organic fruits and vegetables are a simple way to avoid all that, because by law organic farms cannot use synthetic pesticides or genetically modified organisms.
Myself, I try to avoid having processed food, soy, and grains in my diet anyway, so it's quite easy to avoid GMOs.
Here's a link to a popular science article regarding glyphosate being classified as a probable carcinogen:http://www.scientificamerican.com/article/widely-used-herbicide-linked-to-cancer/