Slightly different terms exist among different users (the federal Commerce Dept., for example, nowhere uses the term "Midwest"), but generally you would consider the Northeastern United States to consist of everything between the Mason-Dixon line and New England: the entire states of Maryland, Delaware, Pennsylvania, New York, New Jersey and of course the District of Columbia too.
In a very casual way (say, during a weathercast when visuals take precedence over words), "Northeastern United States" is sometimes used as a blanket term that includes includes New England, but generally New England counts as its own region (i.e., Connecticut, Massachusetts, Rhode Island, Vermont, New Hampshire, Maine; and if I left anyone out I apologize).
Each region has its own historical significance and culture. New England, speaking very generally, is older and more culturally "Anglo" than the Northeastern States to its south. Traditionally the Northeastern States have been the locus of political and economic (and a great deal of industrial) power for the whole nation. That distinction is beginning to slip somewhat given the relative shift in population toward the South and the Southwest (or "Sunbelt" if you must), but note for example how many cultural institutions (Harvard) are in New England and how many commercial institutions (NBC-Universal) are headquartered in some large Northeastern city, usually New York.
Note that the traditional term "Northeastern United States" says very little about geography or climate zones and in fact considerable portions of New York State and Pennsylvania are west of what social-science types like to call "the Appalachian Construct," meaning they are at or beyond the point where American mountains cease to exist 'til the Rockies. It took later to settle the trans-Appalachian west, which is why Pittsburgh sometimes acts cuturally more Midwestern, even though it is in the same state as Philadelphia.
While I'm boring you silly , I'll point out that a term Thomas Jefferson coined for us is frequently very useful: "Eastern Seaboard," which refers to the broad swath of flattish land facing the Atlantic, irrespective of historical regions. TV weatherpeople like to use "Eastern Seaboard," as, say, "Hurricane Whoozits is poised to strike the Eastern Seaboard." Since Florida is a peninsula and semi-tropical, it generally doesn't count, nor do the parts of New England that are craggy all the way to the Atlantic. But "Eastern Seaboard" does embrace Savannah, Charleston, Norfolk, Baltimore, Philadelphia, NYC, Long Island and probably low-lying parts of Connecticut and Massachusetts (I'm not sure about Boston). People who live on (in?) the Eastern Seaboard are certainly aware of that fact!
Using "Eastern Seaboard" as a fairly strict geographical/geological term, to the west of the Eastern Seaboard generally lies the Piedmont, an Italian term the amazing Mr. Jefferson stole that means "foothills"; to the west of the Piedmont begins the whole kit'n'caboodle of the "Appalachian Construct," whether they are Smokies, Poconos, Alleghenies or whatever.
By knowing those three terms, I can pretty much guarantee you an extra ten points on any College Entrance Exam-type exam you may have to take or re-take that has a social-science component. Meanwhile, feel free to impress your friends by using them but for heaven's sake, don't blame me. When you visit Chicago I'll try to define "Midwest," but that is actually trickier than the older historic and geographical regions I've blathered on about above. - al-in-chgo.