Apples, pears, plums, cherries and other stone fruit such as apricots and peaches evolved in central Asia, which has a "continental" climate with hot summers and very cold winters. Although apple-growing in particular is now widespread in many climate zones, nearly all cultivars still require an annual cycle of cold winter weather in order to set blossom and produce fruit each year.
This winter chilling requirement, or minimum chill requirement, is usually defined as the number of hours per year where the temperature should be below about 45F / 7C, but above freezing. Periods when the temperature is substantially below freezing are not thought to be as useful for counting towards chill hours as the period when the temperature is just above freezing.
Most apple varieties have a chill requirement of about 1,000 hours or more, which is readily achieved in the temperate apple-growing regions of the USA, South America, South Africa, Australia, New Zealand, and Europe. Some of the best commercial apple-growing areas are in regions where winter temperatures hover in, rather than below, the chill range - such as Kent, the main apple-growing region of England. If you are in any of these temperate areas then chilling hours are of little relevance to the varieties you can choose to grow.
However if you live in a climate where winter temperatures are not often below this chill threshold, such as southern California or Florida, then you might need to consider growing low-chill varieties. For convenience these are usually classified as needing less than 700 chill hours - and they often seem to prefer hotter summers than other varieties. Medium-chill varieties are those in the range 700 - 1,000 hours.
Note that low-chill and medium-chill varieties can usually be grown successfully in high-chill climates - but high-chill varieties might not do so well in low-chill climates. A good example of this is Gala, which is a medium / low-chill variety, but grows very well in high-chill areas.
Another feature of some low-chill apple varieties is that even in colder climates they have a tendency to retain their leaves long after other varieties have lost theirs.
Israel is at the forefront of the development of apple varieties which have little or no winter chill requirement. Anna, a Golden Delicious style apple, and Ein Shemer, a yellow/green variety, both tolerate climates with 300-400 chilling hours. Dorsett Golden, which was found in the Bahamas, needs less than 100 hours.
As a final word it is worth noting that little research has been done on this subject, and it is perhaps best not to get too precise about the exact number of chill hours, but simply categorise varieties as high, medium, or low chill. In addition, chilling hours is not the only factor in determining whether a particular apple variety will do well in hot climates, but is certainly worth taking into consideration.