"No, I don't need this computer."
Translation:No, no necesito esta computadora.
Yo no necesito … is simply "I don't need …". The English sentence has a "No" at the beginning of the sentence, with the rest of that sentence being added for emphasis or clarification "NO, I do NOT need …”
The Spanish sentence therefore also needs no twice
NO, yo NO necesito …, with the second no negating necesito and is equivalent to "do NOT" which negates "need".
Also, you really should have started a new thread with your question rather than adding it to an existing thread for an unrelated question!
It's like answering a question. So if someone said "do you need a computer?" you would say "no, I don't need a computer," as in this example. Otherwise, speaking without a question first, you would be correct in just saying "I don't need a computer". So that's why there are two "no"'s in the sentence - the first is to answer the question as no, and the second is to change necesito from "I need" to "I don't need". The double negative is bad grammar, but would appear if you said something like "I don't not need the computer," or "yo no no necesito el computadora". Hope that helps!
Double negatives are a "no-no" (pun intended) in the English language, but it's not so in Spanish. As I understand it, in Spanish, the more negatives that are used, the more strongly the negative is being expressed (versus English where a 2nd negative cancels out the 1st negative in a sentence).