Landrace: A genetically-diverse, locally-adapted population.
Variety: Having a distinct appearance from others of the species.
Cultivar: A variety that is selectively inbred (maintained) to keep a consistent appearance.
As examples: My landrace maxima squash can have any colored skin. They can be any size between 5 and 15 pounds. Shape can be banana, round, or oblong. They have to taste great and mature quickly to satisfy the local eaters, and mature in the short growing season. Within my landrace, there are a number of varieties: Orange squash, Green squash, banana squash, round squash etc... Even round orange squash, round green squash, etc.
Landrace containing many varieties:
I grow one squash cultivar. Yellow crookneck. It is always crookneck shaped, and always yellow. I maintain it as a cultivar, and don't let it stray from that type.
It looks from the definitions within that link that varieties generally will be inbred....."true" breeding....and often come from nature.
Cultivars are more the product of human endeavor and will include hybrids (but not always as in Joseph's crookneck), hybrids not being true breeding if they are interpollinated. I guess that makes sense as I've often heard of open-pollinated varieties, but not open-pollinated cultivars, although I suppose such a thing could not be ruled out. Will be interesting to additional discussion....hopefully I've interpreted the artigcle properly.
Edit: I'm seeing a double-posting on this so hopefully one of them can be removed? Also, with regard to when 'variety' is used colloquially, I've rarely heard larger-scale farmers refer, amongst each other, to their corn or sugarbeets as 'cultivars', although they may use the term 'hybrid'. They most often just use the term 'variety' even if a bit generically.
“The most important decision we make is whether we believe we live in a friendly or hostile universe.”― Albert Einstein
Mr. Lofthouse is who I would defer to on this for sure. The only thing that I will add is that I think of cultivars as subsets of varieties. So to expand on Josephs example, there is a variety of summer squash we call yellow crookneck. They are soft skinned squash that mature early in the season and are yellow and crooknecked in shape. Within this there will be a number of cultivars, they will have varying size, color (canary yellow vs easter yellow for example) and they will vary in the exact growth habits/production of the plant.
Don't forget the word grex. Basically a mix a different varieties together that are all growing together but have not interbred. Grex could also be a synonym for a pre-landrae or a proto-landrace if you are developing a locally adapted landrace for your area. A specific variety could be selected out of a landrace.