Some of the information posted here is a bit off, but close to the mark.
Heirloom seed strains are open-pollinated strains that have been around for a long time. Not all open-pollinated strains are heirloom since there are new strains appearing every year, and there isn't a set "age" that a strain must be for it to be called "heirloom." It's really not a clearly-defined term.
Open-pollinated seed strains are those which "breed true", as opposed to hybrid seed which will not. This is comparable to what you can find with chickens -- there are pure breeds available, as well as hybrid types like Golden Comet. If you intend to save seed from year to year, you should get open-pollinated strains -- just like if you intend to "hatch your own" to keep a flock going, you should get pure breeds. If you'd rather get new seed each year, hybrids are fine -- just like if you'd rather buy new chicks every year, hybrids will work for you.
Hybrid seed is created by crossing two pure-breeding lines to give vigor in the first-generation offspring -- which is the hybrid seed you purchase. By crossing two pure-breeding lines, the first-generation offspring will be uniform like a pure-breeding line, but with the benefit of hybrid vigor. This will not be the case with any seed raised from hybrid seed -- there will be variation (not necessarily awful plants, but just not uniform). Hybrid seed is not created by crossing two lines and then breeding further for desired traits and kept isolated and pure -- that is how a new open-pollinated strain is created. With hybrid seed, two separate open-pollinated strains are maintained and inbred further, then crossed once to produce the hybrid seed offered for sale.
Open-pollinated doesn't mean "open air pollinated", which sounds like how some plants (like corn) reproduce -- by sending pollen into the air, to be carried by wind onto the next plant without the help of bees or other pollinators. It means that if planted "out in the open" but isolated from other strains, it will breed true -- it doesn't require human intervention (i.e. crossing two specific strains) for the seed to be produced. Again, think of chicken breeds -- if you have a flock of Rhode Island Reds that doesn't come in contact with any other chickens, that flock will remain Rhode Island Reds with each generation without human intervention (other than maintaining isolation).
GMO stands for Genetically Modified Organism. This is sometimes defined in a way that includes domesticated species which evolved from wild ancestors by selective breeding. Really, it more specifically describes an organism which has genetic material inserted from another species via laboratory techniques rather than cross-breeding. For example, modern wheat is a hybrid of three different species, and remains fertile because of an increase in ploidy (going from a pair of each chromosome to three pairs of each chromosome). But because this occurred naturally (i.e. the plants hybridized on their own, and fertile seed-bearing forms mutated spontaneously without human intervention other than selection of favored forms) without the aid of gene-splicing, modern wheat is not considered GMO. There could, however be a form of GMO wheat if one or more isolated genes from another species are inserted into the genome of a strain of wheat in a laboratory.
It's true that much of the corn produced in the US is GMO (unless stated otherwise), so products made from corn will also thus be GMO (again, unless stated otherwise). However, the only GMO roses out there are those which are the result of attempts to make "blue" roses by inserting genes for blue pigment in the petals, and they are in limited production and rather pricey. Of course, roses grown by florists are very specialized and selected for that industry, and as such are usually very different from the varieties grown in gardens, but that was the result of selective breeding and greenhouse growing rather than being GMO.
