Presbyterians are the descendants of the Scottish Reformed tradition -- "Reformed" meaning "Calvinist," as distinguished from Lutheran, Hussite, Anabaptist, Baptist, etc.
They are presbyterian as distinguished from episcopalian; that is, they are governed by elders (presbyteroi) rather than by bishops (episkopoi).
A quick run-down of US Presbyterians:
Presbyterian Church in the U.S.A. (PCUSA) - by far the largest (over 10,000 congregations), this denomination was formed by the merger of the United Presbyterian Church in the U.S.A and the Presbyterian Church in the United States. Not so Calvinistic. This is where you find the moderates and liberals, though the denomination has its share of conservatives, too. The really hardline evangelical/fundy types have mostly left for other denominations. You can think of them as the "Episcopalians" of Presbyterianism.
Presbyterian Church in America (PCA) - with about 1700 congregations, this is an evangelical (not in the sense of "Lutheran")/fundamentalist group that broke off from the PCUS in the 1970s mostly over women elders, but they also have the full complement of fundy issues. Think of them as the "Southern Baptists" of Presbyterianism. However, the PCA is also Calvinistic and is also the haven of many Dominionists in the tradition of R. J. Rushdoony. In other words, this is where you find most of the really scary-crazy Presbyterians in the U.S.
Cumberland Presbyterian Church, at about 800 congregations, is a mostly Southern, and traditionally rather liberal denomination. With the rise of the Religious Right, the CPC has tended to drift more toward evangelicalism. You can think of them as the United Methodists of Presbyterianism.
Evangelical Presbyterian Church has around 225 congregations. It's a theologically (Calvinist) and socially conservative denomination, though it does ordain women. You can think of them as the "Church of the Nazarene" of Presbyterianism.
Associate Reformed Presybterian Church has about the same number of congregations as the EPC, but fewer members. It's Calvinist and very conservative. Doesn't ordain women, doesn't like gay people or legal abortion, maintains the inerrancy of the Bible. This is the "Lutheran Church - Missouri Synod" of Presybterianism, and it's also the church Billy Graham grew up in.
There are other, smaller Presbyterian denominations in the U.S., of course, but those are the most notable ones.