I am trying to learn the numbers and I see there are two different translations for the number 3 (and other numbers). When should I use one and when should I use the other? Is there a name for the type of number that "san" is and the type of number that "mittsu" is?
There are mainly two types of numbers used in Japanese:
native Japanese numbers (和語数詞 wago sūshi)
1 ひと hito, 2 ふた futa, 3 み mi, 4 よ yo, 5 いつ itsu, 6 む mu, 7 なな nana, 8 や ya, 9 ここの kokono, 10 と to, ...
Chinese-derived numbers (漢語数詞 kango sūshi)
1 いち ichi, 2 に ni, 3 さん san, 4 し shi, 5 ご go, 6 ろく roku, 7 しち shichi, 8 はち hachi, 9 きゅう kyū, 10 じゅう jū, ...
(Some counters may also take English-derived numbers (英語数詞 eigo sūshi) ワン wan "one", ツー tsū "two", スリー surī "three" in low numbers.)
Caveat. Depending on the counter word coming after it, these numbers may slightly change.
The native みっつ mittsu is of the form [number] + [counter word], where the number is the native Japanese み mi that goes with the counter word ～つ (a generic counter for objects) and
み + ～つ → みっつ. (See the above caveat.)
さん san is of the form [number], where the number belongs to the Chinese-derived counting system. It cannot be used to count anything (items, people, ...) by itself.
However, for counting by itself (e.g. during exercise, hide-and-seek, etc.) the Chinese-derived numbers are used (sometimes しち is replaced by なな).
During counting, these are often pronounced rhythmically
イチ【HL】、 ニー【HL】、 サン【HL】、 シー【HL】、 ゴー【HL】、 ロク【HL】、 シチ【HL】、 ハチ【HL】、 キュウ【HHL】、 ジュウ【HHL】
even though they are always written as above.