The Japanese use two sets of numerals for counting, naming quantities, etc. The main set is Chinese in origin, and it is used most of the time. The second set is Japanese in origin, but only counts from 1 to 10 and is only used occasionally as described below.

The Chinese-origin numbers are as follows:

1. ichi
2. ni
3. san
4. shi (yo/yon)
5. go
6. roku
7. shichi (nana)
8. hachi
9. kyuu
10. juu
11. ju-ichi
12. ju-ni
13. ju-san
14. ju-yon
15. ju-go
16. ...
20. ni-ju
21. ni-ju-ichi
...

100. hyaku
101. hyaku-ichi
102. hyaku-ni
...

999. kyuu-hyaku-kyuu-juu-kyuu
1000. sen
1001. sen-ichi
...

You get the idea. For even larger quantities, man indicates 10,000; ju-man indicates 100,000; hyaku-man indicates 1,000,000; and sen-man indicates 10,000,000. Oku indicates one hundred million and combines with ju-, hyaku-, and sen- to form 1 billion, ten billion, and one hundred billion. 1 trillion is denoted by choo, which is the highest number word I can find in my textbooks -- I can't think of too many situations where you would need to specifically refer to more than one thousand trillion things anyways.

The numbers often change slightly (but predictably) in pronunciation as they combine to represent larger quantities. For example, 300 is sanbyaku rather than "sanhyaku". Likewise, 800 is happyaku rather than "hachihyaku". 8000 is pronounced hassen, 600 is pronounced roppyaku, and so on. It's not as complicated as it seems, once you get used to it.

These Chinese-origin numbers are used for the vast majority of counting and denoting quantities, including times of day and ordinals (1st, 2nd, 3rd, ...). For counting objects or describing a quantity of things, such as three pieces of paper, a counter is suffixed to the number. This is like saying "three head of cattle" or "seven slices of bread" in English -- the difference is, the counter cannot be omitted in Japanese. (See "Japanese Counters" for a short list of common Japanese counters, or individual counters by name).

To generate ordinals, simply suffix "ban" to the Chinese number:

Ichiban, niban, sanban.
Number one, number two, number three.

To express "first", "second", etc, use "banme":

Yonbanme, gobanme, rokubanme.
Fourth, fifth, sixth.

The Japanese-origin numbers are as follows:

1. hito
2. futa
3. mi
4. yo
5. itsu
6. mu
7. nana
8. ya
9. kokono
10. too
These numbers are used in very certain circumstances that are difficult to catalogue. Of particular interest to foreigners is that these numerals are used with the counter "tsu", which denotes miscellaneous things or is used when the proper counter is unknown. For example:

Piiza o futatsu o-negai shimasu.