Long and short numeric scales
To cut a long story short, the difference between the long scale and the short scale is that:
In the short scale, every new word greater than a million is one thousand times bigger than the previous term (the digits are grouped by three).
For example, one million is 106 and one billion is 109. Next scale word is one trillion, which is 1012.
In the long scale, every new word greater than a million is one million times bigger than the previous term (the digits are grouped by six).
For example, one million is 106, one thousand million is 109 and one billion is then 1012. One trillion jumps to the 1018 position, as the previous scale, 1015, is occupied by another name matching one thousand billion, and the two naming series go on alternatively.
A European-centric vision
Historically, the long scale was used in France from the turn of the 15th century, spread out in Europe until the 17th century when the short scale was devised. The short scale was now in favor. In the meantime, the world was “discovered” by Europeans who spread out the short scale in their new colonies (and sometimes the long scale too, which was then replaced by the short one).
After some back and forth between the two scales, the situation can be summed up like that: European countries are now using the long scale (with the exception of the United Kingdom), whereas some previous colonies of the European empires kept the short scale system (Brazil, United States of America), and others kept the long scale (all the Spanish-speaking countries, with the exception of Puerto Rico).
Different cultures, different scales
Grouping numbers by three or six digits is not the only way to name big numbers. Modern Chinese groups them by myriads (or groups of four) (亿 is 108, 兆 1012), as well as Japanese (万 is 104, 億 108) and Korean (만 is 104, 억 108). Hindi groups them by two after 1,000 (सहस्र is 103, लाख 105, करोड़ 107), traditional Tongan has a special name for intermediate powers of 10 (mano is 104, kilu 105)…
The beauty of languages is that they are full of possibilities. Even in such a restricted area as how to name big numbers, the differences in counting and naming are quite awesome, opening up new windows into the cultures supported by each language.