"Whig" comes from whiggamore, a dialect word referring to a Scots bumpkin. It was what Cavaliers called Roundheads. "Tory" comes from Irish toraidh, meaning an outlaw or renegade. It was what the Williamites called the Pretender's sympathizers.
By the time of our own Revolution, Tory came to describe the backers of George III, and then the aristocratic party generally. Those who backed the "Rights of Englishmen" proudly called themselves Whigs -- which was the original label used by Thomas Jefferson and others, before "Patriot" came into use. Later on, the opponents of Andrew Jackson resurrected the Whig label.
Whiggery is also called "classical liberalism." It's the viewpoint the Founders enshrined in the Declaration of Independence, the Constitution, and the Bill of Rights. But against the grasping ideology of Progressivism, it has had to take up residence in Tory territory. Mostly the Whig tradition in America is now what we call "conservative" in order to avoid confusion. "Liberal" really doesn't describe much any more, since Progressives claim the label, even though they are a pretty illiberal bunch. I don't know anybody who's an old-fashioned aristo-snoot any more, so if there are still Tories about, I couldn't point you to one.
Instead, Whiggery has separated into various forms of conservatism/classical liberalism. There are the Libertarians, the defense hawks (Neo-Cons), social-values voters, and constitutionalists. Several of these identify as Tea Partiers, since the issue of the day is restraining the advance of government and most conservatives think that a desirable thing. The essence of Whiggery was always freedom.
The Republican Party started out as a coalition of Abolitionists and Free-Soilers. One group wanted to eliminate slavery and one wanted to restrain its expansion. While never losing its focus on civil rights (Republicans voted overwhelmingly for the Civil Rights Acts in the '60s, without whom they could not have passed the Congress), the GOP also became in the latter 19th Century the patron of Small Business (not Big Business, which backs any party that can achieve and keep power). As a party dedicated to traditional rights and responsibilities, the GOP is the home of Whiggery today, mixed with other points of view on the right side of the aisle.
So, we have a Center-Right party (the GOP) and a Radical Left party (the Democrats) in America today. Various small parties raise particular issues which are in time claimed by the two major parties. With our winner-take-all electoral system, we really can't sustain more than two major parties at any given time. Lots of people gripe that the Republicans aren't very pure in their conservatism. What they forget is that conservatism is a broad spectrum these days, which includes the center. Only the far left is pure in their ideology, and they polarize everything around their agenda, which one cannot be indifferent to.