Need vs. do need.
How could I understand the difference between those two ways?

You need - You do need.

Question

Need vs. do need.
How could I understand the difference between those two ways?

You need - You do need.

Answer

Do means "not not"

A: "I need this"
B:  "Wrong!  You don't need it."
A: "No, YOU'RE wrong!  I DO need it!"

Some people don't need jackets in cool weather, but other people do need them.

Answer

Saying 'I do...(+ verb)' is just to emphasize it, like 'ok, maybe I don't... but I DO...'   
That's all.   
It's for any verb, it's not related to 'need'

'You can do the same thing with 'am' and adjectives, and so on)

Answer

I need = Yo necesito
I do need = Yo sí necesito   ("Creo que no necesitas" ...  "Estás equivocado, yo sí necesito")

Answer

If I understand you are asking the difference between:

I need to do my homework.

I do need to do my homework.

The first one is a more proper sounding English sentence for writing without emphasis. The second one is often spoken by natives when they want to emphasize the answer to the question "Do you need to do your homework?" is a positive answer and not "I don't need to do my homework." This often leads to funny sentences like "I do do my homework when I have time" which people will laugh at. I never see it written this way however.

Answer

You mean;  'What is the difference between these?'

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