Cultivars (short for cultivated varieties) are plants you buy that have been propagated not from seed, but rather vegetatively (e.g., via stem cuttings). With this method of propagation, you can be sure that the offspring will retain the characteristics of the parents for just the one generation. I.e., plants grown from the seeds of cultivars may disappoint you, failing to stay true to form.
When the full scientific name for a particular plant cultivar is given, the part of the name that indicates the cultivar itself follows the genus name and the species name and is set off by single quotation marks. By referring to such plants in this way, we're able to be more specific about them than if we restricted ourselves to noting genus and species.
By contrast, a "variety" (sometimes abbreviated var.) arises naturally in the plant kingdom, and plants grown from its seeds will typically come out true to type. If you remember that "cultivar" stands for "cultivated variety," you'll have no problem remembering the difference between the two. Whereas a plain old "variety" is a natural phenomenon, a cultivated variety has been produced via human intervention -- just as a cultivated piece of land has been altered to suit human needs, as opposed to one left in its natural state. In fact, "cultivated" derives from the Latin word meaning "to till (the soil)."
When a variety is named, it appears differently than a cultivar name does. Rather than being presented in single quotes, it is italicized and in lower case -- just like the species name, which it follows.