"Dw i eisiau mynd i Efrog Newydd."

Translation: I want to go to New York.

[RalphOsmond]

it is not sinking in, can someone please explain when I use "dw i" as opposed to "dw i'n"

[ibisc]

'n/yn is needed to make the link between dw i, rwyt ti, etc and a verb-noun such as mynd, dod, gweld, etc:

• Dw i'n gweld y ci. - I see the dog.
• Rwyt ti'n mynd i Aber - You are going to Aber.

See https://preview.duolingo.com/skill/cy/Present-1/tips-and-notes. Also see the summary of yn here - https://forum.duolingo.com/comment/13844144

eisiau is not a verb-noun, and the pattern dw i eisiau, etc is an exceptional one in Welsh - the 'n/yn is not used. This is explained in the course notes here - https://www.duolingo.com/skill/cy/Wanting/tips-and-notes

[The reason for this is that there is a more formal form that is occasionally used where there is another word between dw i and eisiau, but that word has been dropped in colloquial use.]