I want to tattoo the word half demon but in japanese. Do you spell it hanyo, han'yo or hanyou? I saw all of them? Which one is correct?
There are multiple systems of romanization and various conventions.
One important point relevant to this word is that 'n' (hiragana: ん) is unique in Japanese in that it is the only non-vowel-containing "sound unit", and that can cause ambiguities.
半妖 is spelled はんよう in hiragana, which represent ha-n-yo-u. But if you write just hanyou, it could be misread as ha-nyo-u (or in hiragana はにょう). This doesn't make it 'wrong', but there is potential for confusion. (Luckily there don't seem to be any common words with this spelling so a reader would hopefully figure out what you intended, but, in general, there is definitely potential for confusion when romanizing "n" without the apostrophe when it is followed by vowels or glides).
The apostrophe after the n is used to "close" it (replaces the vowel) to eliminate the ambiguity... it is optional, but much safer.
For representing the "reading" of a Japanese word containing kanji, the safest approach is usually to directly use hiragana, eg はんよう. If you insist on using romaji, "han'you" is safer because it is unambiguous. The long "ou" can also be written "ō", or reduced to just "o" (but the later is potentially ambiguous with はんよ [ha-n-yo]).
Also note that Japanese has many homophones (same-sounding words). So there are other words that are read はんよう, but don't have anything to do with half-demons. For example 汎用, meaning "general purpose". If this is for a tattoo, I would also bring in the kanji spelling itself so that no confusion occurs.
Finally, this is very tangential, but: if you think you might go to Japan some day, please think that tattoo through. Tattoos are commonly associated with organized crime here, and many public baths still refuse service to people with visible tattoos (this is slowly changing, but, .. slowly).