A cultivar is a cultivated variant of a species.
Cultivars may either occur because of cross-pollination or by natural variation in plant characteristics.
Cross-pollination may be accidental (e.g. bees pollinating different plants in proximity) or deliberate (e.g. by people hand-pollinating two different plants).
Cultivars may also happen if buds or cuttings are grafted from one kind of plant to another.
Cultivars may not reproduce true from seed (e.g. seeds from a Weeping Cherry, if planted, may not produce another Weeping Cherry).
Cultivars may arise in the wild, but may be the results of natural variation. The cultivar is marketed for a certain trait, e.g. flower color (Spring Snow Crabapple, scientific name Malus 'Soring Snow'); tree size or shape (Weeping Norway Spruce, scientific name Picea abies 'Pendula'); disease resistance (several cultivars of Dutch Elm, scientific name Ulmus x hollandica), which are resistant to the Dutch Elm killer Dutch Elm Disease.
There must be an upside for plant breeders doing this work: Cultivars are given "plant patents", which are like other patents: the plant breeder has rights to the cultivar, and either someone else is forbidden to propagate for sale the cultivar, or the plant breeder gets a royalty. For example, U. S. Plant Patent PP12,673 for 'Long,' a cultivar of the hybrid Quercus x warei.
Plant patent lives vary, but 25 to 30 years is the common length.
The standard notation for cultivars is single quotes, and the cultivar name is non-italicized.