This is a very difficult but very important question, and unfortunately no one has a clear answer to it at this point. It is important to remember that virtually all food crops have been extensively modified for centuries by selective breeding and hybridization, a process which was naturally limited by biology and which is accepted as safe. However, the more recent technologies which allow the creation of Genetically Modified Organisms (GMOs) have allowed scientists to change organisms much more quickly, and to transfer genes between totally different organisms, even between biological kingdoms. When DNA is moved into an organism, that organism becomes capable of making proteins that it could not produce naturally. These proteins can cause a wide variety of effects.
There are some positive things to be said about using GMO technology to produce food. Crops can be created which produce good yields in less than optimal conditions. We can grow crops which have larger quantities of certain vitamins or other phytochemicals in them. Much GMO technology has been directed toward lowering the dependence of farmers on pesticides, which is obviously a good thing.
One major fear that people have about eating GM foods is the possibility of developing sensitivities or allergies to the proteins that the transferred DNA produces. While this has not been proven to occur so far, the possibility exists. Another concern is how one's body will react to the transferred DNA itself when a GM food is consumed. There have been some studies that indicate that mammal cells can uptake functional DNA from foods that have been ingested, and this has created concerns about what could happen, particularly as some GMO crops are modified specifically for antibiotic resistance.
The bottom line is, no one knows. Virtually every American has consumed considerable quantities of GMO food already, with no obvious consequences. At this point the debate, at least in the US, centers more on labeling and people's right to know what is in their food. GMO foods are here in our diets, but no one yet knows the long term consequences.