Well, to start off, there are two basic sets of numbers, from one to ten.
The more frequently used (and typically the first one learned in class):
ichi, ni, san, shi, go, roku, shichi, hatchi, kyuu, juu.
The other basic system that gets used with general objects and a few other items:
hitotsu, futatsu, mittsu, yotsu, itsutsu, mutsu, nanatsu、yatsu, kokonotsu, too.
Then, just to confuse things, people frequently use yon instead of shi for 4, and nana instead of shichi for 7.
Then you have the counter words, which change depending on what kind of thing you’re counting. According to my teacher back in college, some professor in Japan did some research on these and found that there were over 120 different counter words, although the average person doesn’t use or even know anywhere near that many (there’s a popular comedy show on TV here that regularly has a segment where the cast members and a guest play a game where each person names an object and the next person has to name the correct counter. If you make a mistake a sumo wrestler comes out and beats the crap out of you).
These counter words can also change slightly depending on the number, to make the sound flow more smoothly. For example:
Cars (dai): ichi-dai, ni-dai, san-dai, yon-dai, go-dai, roku-dai, nana-dai, hachi-dai, kyuu-dai, juu-dai.
Bottles (hon): ip-pon, ni-hon, san-bon, yon-hon, go-hon, rop-pon, nana-hon, hap-pon, kyuu-hon, jup-pon.