As a horticulturist, it is important that I use the right terms the right way (at least most of the time). Variety and cultivar are two terms often abused by gardeners and horticulturists. What's the difference, you ask? Quite a lot.
Both are part of the scientific name. Both appear after the specific epithet (second term in a scientific name). Both refer to some unique characteristic of a plant. However, this is where many of the similarities end.
Varieties often occur in nature and most varieties are true to type. That means the seedlings grown from a variety will also have the same unique characteristic of the parent plant. For example, there is a white flowering redbud that was found in nature. Its scientific name is Cercis canadensis var. alba. The varietal term "alba" means white. If you were to germinate seed from this variety, most, if not all would also be white flowering.
Cultivars are not necessarily true to type. In fact cultivar means "cultivated variety." Therefore, a cultivar was selected and cultivated by humans. Some cultivars originate as sports or mutations on plants. Other cultivars could be hybrids of two plants. To propagate true-to-type clones, many cultivars must be propagated vegetatively through cuttings, grafting, and even tissue culture. Propagation by seed usually produces something different than the parent plant.
Varieties and cultivars also have differently naming conventions. A variety is always written in lower case and italicized. It also often has the abbreviation "var." for variety preceding it. The first letter of a cultivar is capitalized and the term is never italicized. Cultivars are also surrounded by single quotation marks (never double quotation marks) or preceded by the abbreviation "cv.". For an example of a cultivar of redbud, consider Cercis canadensis 'Forest Pansy' (or Cercis canadensis cv. Forest Pansy) which has attractive dark purple spring foliage and pinkish-purple flowers.
Can a plant have both a variety and a cultivar? Sure. One good example is Sunburst Honeylocust. Its scientific name is Gleditsia triacanthos var. inermis 'Sunburst'. The term "inermis" means without thorns and "Sunburst" refers to the bright golden spring leaf color.
In today's world of horticulture, cultivars are planted and used more than varieties. Yet we often still refer to a type of plant species as a variety instead of what is actually is a cultivar. Let's kick off the New Year by being more accurate and start using the term cultivar.