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What does the "N" in N-nitrosoamine mean/stand for? Also, if the functional group is connected to a benzene ring what do I call it?

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The "N" means that the nitroso group ($\ce{-NO}$) is attached to the amine nitrogen.

So, for example, in N-nitrosomorpholine the nitroso group is attached to the nitrogen atom of the saturated, heterocyclic morpholine ring:

N-nitrosomorpholine

You could never have an "N-nitroso" group attached to a plain benzene ring, since there is no nitrogen atom in a benzene ring for the "N-" prefix to refer to.

In general, though, you would name aromatic nitroso compounds in the same way as you would those with any other substituent like 'chloro-' or 'nitro-'. For example, the following compound would is named nitrosobenzene:

Nitrosobenzene

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A nitroso group is a nitrogen double bonded to an oxygen (think carbonyl). When that nitroso group is attached to an amine, it is an N-nitrosoamine. We add the N- designation so we know that the nitroso group is bonded to the nitrogen of the amino group as opposed to somewhere else on what is presumably an organic molecule.

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