A landrace is very nearly the same thing as a heirloom plant; as a matter of fact, they may indeed be the same thing, depending on which definitions you use.
Landraces are varieties of plants that have become adapted to a specific locale. They may be optimized for growing in the local soil, be hardier than usual in the face of local weather conditions (such as flooding, freezing, wind, etc.), or be resistant to local pests.
Once these traits become recognized and intentionally selected for by the farmer, the landrace will start to develop into a more clearly defined phenotype, and will be called a cultivar. If this process continues long enough, a new species will develop, resulting in a new cultigen. However, these terms have some overlap, and any cultigen that has not yet been formally recognized as such may be called a landrace.
Landraces can begin to develop in just a few generations, particularly if the local population starts with a number of varieties of plants that can crossbreed. Any locale where seeds are saved and reused from year to year will naturally start to develop landraces; hence, most developing nations will have diverse and thriving collections of landraces for any given locale. Developed nations, on the other hand, often rely on seed bought from companies specializing in carefully bred or genetically modified stock, resulting in higher yields but lower genetic diversity.
The difference between landraces and heirloom plants is primarily a matter of marketing; heirloom varieties may be landraces or cultivars, but essentially, heirlooms are called heirlooms because that makes them sell better. And this is fine. There is, however, one important advantage to using the term landrace, in that it emphasizes that the plant is evolved to thrive in a very specific local environment. If it is removed from its native environment it will start to develop into new landrace. In other words, landraces are not something to be preserved as-is, but to be encouraged to change and develop.
The term landrace can also be used when talking about animals; in this case it is one step below breed, although there are some breeds that have been called 'landrace', as in the Danish Landrace pig.
The word landrace comes, simply enough, from the words 'land' and 'race'.