There are two main English-language systems of number names: the Americann system (also used in Canada) and the European system (used in most Germanic and Romance languages). The European system is the older one and is also called the British system, because it was used in Great Britain at one point. Britain and Australia have recently largely switched to the American system. The American system was originally invented by the French, adopted by the United States, and then abandoned by France.
The principal difference between the two systems is the meaning of the word "billion" and its multiples: in the European system a billion is a million million (10^{12}), a trillion is a million billion (10^{18}), and so forth; a thousand million (10^{9}) is called a milliard, a thousand billion (10^{15}) a billiard etc. In the American system a billion is a thousand million (10^{9}), a trillion is a thousand billion (10^{12}), and so forth. See more below.
In the European system (see Chuquet!), each number is a power of a million, i.e. (n)illion = (one million)^{n} = 10^{n×6}.
one billion = (one million)^{2} or 10^{12};
one trillion = (one million)^{3} or 10^{18};
one quadrillion = (one million)^{4} or 10^{24};
one quintillion = (one million)^{5} or 10^{30};
one sextillion = (one million)^{6} or 10^{36};
one septillion = (one million)^{7} or 10^{42};
one octillion = (one million)^{8} or 10^{48};
one nonillion = (one million)^{9} or 10^{54};
one decillion = (one million)^{10} or 10^{60};
one undecillion = (one million)^{11} or 10^{66};
one duodecillion = (one million)^{12} or 10^{72};
one tredecillion = (one million)^{13} or 10^{78};
one quattuordecillion = (one million)^{14} or 10^{84};
one quindecillion = (one million)^{15} or 10^{90};
one sexdecillion = (one million)^{16} or 10^{96};
one septendecillion = (one million)^{17} or 10^{102};
one octodecillion = (one million)^{18} or 10^{108};
one novemdecillion = (one million)^{19} or 10^{114};
one vigintillion = (one million)^{20} or 10^{120};
...
one centillion = (one million)^{100} or 10 ^{600}.
In the American system, each number is a power of a thousand (the prefix and the number of power will match only when the first 1000 is factored out), i.e. (n)illion = 1000 × 1000^{n} or 10^{(n+1)×3}.
one billion = 1000 * 1000^{2} or 10^{9};
one trillion = 1000 * 1000^{3} or 10^{12};
one quadrillion = 1000 * 1000^{4} or 10^{15};
one quintillion = 1000 * 1000^{5} or 10^{18};
one sextillion = 1000 * 1000^{6} or 10^{21};
one septillion = 1000 * 1000^{7} or 10^{24};
one octillion = 1000 * 1000^{8} or 10^{27};
one nonillion = 1000 * 1000^{9} or 10^{30};
one decillion = 1000 * 1000^{10} or 10^{33};
one undecillion = 1000 * 1000^{11} or 10^{36};
one duodecillion = 1000 * 1000^{12} or 10^{39};
one tredecillion = 1000 * 1000^{13} or 10^{42};
one quattuordecillion = 1000 * 1000^{14} or 10^{45};
one quindecillion = 1000 * 1000^{15} or 10^{48};
one sexdecillion = 1000 * 1000^{16} or 10^{51};
one septendecillion = 1000 * 1000^{17} or 10^{54};
one octodecillion = 1000 * 1000^{18} or 10^{57};
one novemdecillion = 1000 * 1000^{19} or 10^{60};
one vigintillion = 1000 * 1000^{20} or 10^{63};
...
one centillion = 1000 * 1000^{100} or 10 ^{303}.
For more information on prefixes, see SI prefix. See also Cardinal number and Natural number
NOTE: Most of these number names are not used in ordinary writing. The "-illion"s above "quintillion" are almost nowhere to be seen, except perhaps in lists of number names such as this one. Larger numbers are referred to as powers of ten using scientific notation, for instance, as "10^{25}" (read "ten to the twenty-fifth"), etc. "Googol", however, is not uncommonly used for rough approximations.
Zillion and gazillion are fictitious very large numbers. The terms 'a (ga)zillion' and '(ga)zillions' are used informally, meaning very many, as in the Zillions of Games software product.