Most plants and animals of dietary consequence to us are diploids. That means that, like humans, they each carry two sets of their genetic material in their cells. When procreating, each parent has to make sex cells which carry only one set of genetic material - so that the progeny doesn't get a doubled set. This process of halving the DNA content is called meiosis, and is a system full of checks and balances. Should anything go awry, in most cases, the resulting sperm or egg cell is infertile.
So, if you want to make a seedless plant, you have to set up a situation where this process doesn't go as planned. One common strategy is triploidy.
|The little black specks are the abortive seeds.|
Not to say that triploidy can only be applied to plants. One of the animals that we take advantage of triploidy in: oysters. Unable to make viable sex cells, the decreased gonad sizes mean that triploid oysters become larger and devote more their resources making, well, themselves. Think of them as molecular eunuchs.