Yesterday I was talking with someone who is newer to gardening than I am and these terms came up and the person commented on needing clarification on what these meant. In horticulture, a hybrid is a plant that was created by cross-pollinating specific plants. Plant hybridizers do this to improve certain aspects of plants that make them more appealing. Usually the aim is for bigger flowers, stronger resistance to diseases, bigger fruit or better colors. Most of what is available in your local grocery store are hybrids and saving seeds from those fruits will not usually produce a plant that will grow fruit that taste like what you ate.
Heirlooms, just like the name implies have been saved and grown from seed by generations of gardeners and farmers. Probably the most classic example of heirloom varieties are heirloom tomatoes which are sought after for their superior flavor, color or texture. You can find heirlooms at places like farmers markets where they have a large following. These seeds can be saved and grown again to produce a fruit just like the one you ate. Many heirloom varieties are at risk of being lost and gardeners who like to preserve seed diversity go out of their way collect or buy heirloom seeds.
Genetically engineered plants are plants that have had their DNA modified. I'll refrain from trying to explain something that is above my pay grade. Instead I'll direct you to the Wikipedia page for GMOs. I will say that I'm not comfortable with the idea of genetically engineered plants growing in my garden even if they are just being grown for ornamental purposes.
Open pollinated is another term you'll encounter in seed catalogs and if you trade seeds with reputable gardeners. Open pollinated plants are plants that have been pollinated by insects, like bees, or the wind. Generally the seeds will grow a plant with the characteristic that have been promised but you may find some variability. You can't stop bees from collecting and distributing pollen and you can't stop the wind from carrying pollen.