Ah, what a question. You should consider coming to the even I am planning for September :-) (See CommunityBuzz for that)
Anyway, having taken classes in both the History of Newark, as well as the History of the American City, I will attempt to sum things up for you all. Appologies for any rambling.
Applying "normal" terminology for the New York City metropolitan area will make anyone go crazy. For example, "Downtown" New York is not the central business district, as "downtown" Chicago, Boston, or Los Angeles would be, it is actually just the lower portion of Manhattan.
In the same boat, New York, largely due to geographic limitations developed as an incredibly dense city, unlike other cities which are spread out, and continue to annex additional lands under the city limits. For comparison, New York is roughly 165 square miles of land.. Chicago is 234, and Los Angeles is 568.
If the Los Angeles City limits were overlaid over the New York area, the six largest cities in New Jersey, would be included in those boundaries (Newark included). New Jersey is in fact the only State to not have a MSA (Metropolitan Statistical Area) of its own. North Jersey is part of the Greater New York MSA, and South Jersey is part of the Greater Philadelphia MSA.
As far as what makes a suburb a suburb, you can look at a suburb which has an economic, social, or demographic tie to a larger city. If you consider the fact that people regularly commute up to two hours each way into New York, from Eastern Pennsylvania, Southern Connecticut, Southern New York State, and a good Portion of New Jersey, New York City's suburban reach is gigantic.
Newark is indeed a suburb of New York, but has its own suburbs, such as Belleville, Bloomfield, Irvington, East/West/"Regular" Orange, and Elizabeth, all of which are also suburbs of New York City.
Other such examples of Cities which are also New York Suburbs in the area are Trenton, New Brunswick, Paterson, Elizabeth in New Jersey, Poughkeepsie, Middletown, Newburgh, Yonkers in New York, Stanford and Greenwich in Connecticut, and even Allentown and Scranton in Pennsylvania.
I hope I have managed to not totally lose you all, and that this makes some sense to you.