In 1793-94, there were a number of clubs that were formed in the United States to celebrate the French Revolution, promote democratic values, and fight against the creeping influence of aristocracy in American society. The clubs went by various names - "Democratic," "Republican," "True Republican," "Constitutional," etc. - but historians generally refer to them as "Democratic-Republican Societies".
Around the same time, supporters of the conservative faction of the government - led by Alexander Hamilton and John Adams - started calling themselves "Federalists", and described their opponents as "Republicans", "Democrats", or "Jeffersonians", after their leader, Thomas Jefferson.
By 1794-95, these two factions began organizing themselves in local and state elections, seeking to win offices and patronage appointments for their followers. As they contested elections, the labels of "Federalist", and "Republican" - or "Democratic-Republican" - began to stick. As Jefferson wrote in a February 12, 1798 letter:
Two political Sects have arisen within the U. S. the one believing that the executive is the branch of our government which the most needs support; the other that like the analogous branch in the English Government, it is already too strong for the republican parts of the Constitution; and therefore in equivocal cases they incline to the legislative powers: the former of these are called federalists, sometimes aristocrats or monocrats, and sometimes tories, after the corresponding sect in the English Government of exactly the same definition: the latter are stiled republicans, whigs, jacobins, anarchists, disorganizers, etc. these terms are in familiar use with most persons.
The Federalists faded away after the War of 1812, but the organization and name of the Democratic-Republicans continued during the Era of Good Feelings (1817-25), which was characterized by a lack of partisanship in national politics.
But the first word of the party's label, "Democratic", was revived by supporters of Andrew Jackson in the hotly contested election of the 1828. By the 1830s, supporters of Jackson - organized by party manager Martin Van Buren - regularly started calling themselves "Democrats", reflecting both the populist style of the Jacksonian party, but also their connection to the Jeffersonian tradition.
Later, in the 1850s, another Republican Party arose in the ashes of the Whigs (who had been organized in opposition to Jackson in the 1830s). Their name was suggested through a series of exchanges between newspaper publisher Horace Greeley and lawyer Alvan E. Bovay in 1852-54, as it became clear that the Whig Party was falling apart under the weight of the slavery issue.
Bovay later explained the reasons why the name was chosen:
He thought that parties should have simple names that symbolized their ideology. Republican came from the Latin "res publica" - common weal. This distinguished them from the Democrats, whose credo was - as Bovay put it - "I am as good as you," whereas the Republicans believed "You are as good as I".
He wanted it to be a simple word and not a compound, as was the case with the "Free-Soil" Party.
He wanted a word that could be used as both a noun and an adjective.
He liked the fact that the same word had been used by the political party of Thomas Jefferson, symbolizing a link with the country's ideological past.
He thought the name would appeal to immigrant voters, and would draw them away from the Democratic Party.
(Source for Bovay: http://content.wisconsinhistory.org/cdm/compoundobject/collection/tp/id/46379/show/46363)