Open-pollinated varieties have been developed through selective breeding (choosing the individuals with the most desirable traits) over many generations until the most desirable traits are consistently present. This means that they will come true from seed, or that plants growing from their seed will be more or less the same as the parent.
Heirlooms are open-pollinated varieties that are usually at least fifty years old, although some people only consider varieties with a history of development by home gardeners to be heirlooms. Growing heirlooms can be an important part of maintaining genetic diversity, which gives future plant breeders a broader range of characteristics, including disease resistance, that they can draw from. This doesn't mean that heirlooms are disease resistant; one of the downsides of heirlooms is that they are often quite susceptible to modern diseases, and prone to other problems as well. Heirlooms are often very well suited to the location where they were originally grown, and sometimes perform poorly in other conditions.
Because open-pollinated varieties are often newer introductions than heirlooms, they are usually more productive, disease resistant and uniform than heirlooms, but they are still a good choice for gardeners who save seeds for planting the next year.
Hybrids are usually a cross between two open-pollinated varieties. Two individuals that have been selectively bred for certain traits are crossed, producing offspring with the most desirable traits and usually very productive (hybrid vigour). The offspring are generally very uniform, but will not reproduce true to type. Hybrids are very popular with commercial growers, as the produce is usually very uniform, disease-resistant and productive.
Some gardeners dislike hybrids and feel that they are unnatural, however, hybrids often occur in nature. The real problem with growing hybrids is simply that because less varieties are being grown, genetic diversity is lost. Growing hybrids also makes the gardener dependant on the seed company for seeds every year, but sometimes the hybrid variety is so much more productive or reliable that it is worth it.