With the fading of the Federalist party in many areas and the lack of organized political "parties" in the modern-sense, several candidates for election were often described in terms of their support for a single candidate.
New York 1806-08: Morgan Lewis was the Governor of New York from 1804-07. He was elected in 1804 with the support of DeWitt Clinton, but after their split, those supporters of Lewis would be described as such in many newspapers.
Pennsylvania 1805, 1811: With the split in the Republican Party in Pennsylvania in 1805 (See
"A strong and aggressive Federalist Party had contributed much to the Republican victory in Pennsylvania in 1799. It had forged Republican unity and, by its excesses, had added large numbers to the ranks of its opponents. After the election of 1800 Federalism in the State declined precipitately; and within two years John Quincy Adams was to describe it as 'so completely palsied, that scarcely a trace of it is to be discovered except in here and there a newspaper edited by New England men.' (ft: John Quincy Adams to Rufus King, October 8, 1802 in Charles R. King (ed.) The Life and Correspondence of Rufus King, 6 vols. (New York, 1894-1900), IV, 176.) Gratifying as such a metamorphosis must have been to the Republicans, it was not without its cost. The virtual disappearance of Federalism weakened the compulsion for unity and gave play to Republican differences on measures and men which by 1802 had resulted in a number of local divisions in the party." (The Keystone in the Democratic Arch: Pennsylvania Politics, 1800-1816. Sanford W. Higginbotham. 1952. p 25)
"Whatever the true explanation of these intricate political maneuvers, the  senatorial election had disclosed division in the Republican ranks. The party had begun a new era in its history." (Higginbotham, p 34)
Pennsylvania 1808, 1813-14: Michael Leib was elected in 1808 as a U.S. Senator from Pennsylvania. In that same year, Thomas Humphreys, a candidate for the Bank Director of the Bank of Philadelphia was described by The True American as "Leib's candidate." Later, in 1813 and 1814, factions would develop in the Republican Party and many candidates for elections in these two years would be described by The Democratic Press (Philadelphia) as Leibites.
Much of the division in Pennsylvania, and specifically, Philadelphia politics stemmed from those supporters of Simon Snyder, who was the Speaker of the House, and later Governor, and Michael Leib, whose power over the Republican party lead to the Constitutionalists, those supporters of then governor Thomas McKean, to split away from the Republican Party in Pennsylvania as a whole and form their own party for several years, from 1805-1808.
"The new era was dominated by two themes. The first of these was the national issue of supporting the administration's foreign policy, including the War of 1812. Party lines were sharply drawn, and a strong Federalist minority took an active part in politics. The second was the bitter feud between the Leib-Duane faction and the followers of Snyder. This persisted in full rancor throughout the period and was only partially subdued by the compulsion for the Democratic unity exerted by the War of 1812." (Higginbotham, p 177
"The second period, which ended with the election of 1808, was characterized by two main questions - whether the Federalist-Quid coalition was to form the basis of a permanent new party; and whether the city Democrats, led by Leib and Duane, or the country Democrats, controlled by the adherents of Snyder, should dominate the party. The growing importance of foreign relations arising out of American neutrality in the Napoleonic wars settled the first question in the negative and forced the postponement of a decision of the second. In the face of a resurgent Federalism, Pennsylvania Republicans suppressed their differences and united in a successful support of Snyder, Madison, and the embargo. Foreign affairs continued to be important for the next three years; but congressional vacillation and the relaxing of Federalist efforts within the State permitted the Snyderites and the Duane-Leib faction, now known as the Old School, to fight out their battle for control of the party. The Olmsted affair offered the occasion, and for a time it appeared that the Old School might be victorious. However, its own intemperate violence and political blundering redounded to the benefit of the Governor adn his adherents; and by 1811 Duane had forsworn State politics, and the Old School consisted only of Leib and a few hangers-on. The Snyderites not only dominated the State as a whole, but, acting through Binns, had achieved supremacy in Philadelphia." (Higginbotham, p 328-329.
Pennsylvania 1819 Speaker of the House: Joseph Lawrence is listed as a Findlayite. In the same election, Rees Hill is listed as a Binnsman by the American Republican of December 14, 1819 and as a Binnite by the Crawford Weekly Messenger (Meadville) of December 17, 1819. The Village Record of December 15, 1819 lists Lawrence as an Administration candidate and Hill as an Anti-Administration candidate.
Republican splinter parties
New Jersey 1820: Several newspapers, including the Elizabeth-Town Gazette and the True American (Philadelphia) listed a separate ticket of dissident Republicans for the U.S. House of Representatives race in New Jersey in 1820, referred to as the "Anti-Caucus" ticket. Nominations for At Large candidates on a state wide level could often cause problems. Rotation of candidates, or lack thereof, from different regions/counties would sometimes cause dissension, and occasionally regional candidates, often an incumbent who had been dropped from the list, would be set up in opposition. As the Federalist Party declined, the process of country meetings, conventions and the Legislative caucus to nominate candidates came under increased criticism and with less party competition the idea of a more open and balanced method of selecting candidates was becoming a political issue.
Adamite / Crawford:
While many tickets would grow up around support for one person (such as Clintonians in New York or Snyderites in Pennsylvania), the affiliations of many candidates in various elections in 1823 and 1824 were based around which candidate for President in 1824 the individual candidate was supporting. While those supporters of Andrew Jackson would become the mainstream part of the Republican Party as it transitioned into the Democratic Party, there were also the followers of John Quincy Adams, many of whom would soon form the basis for, first the National Republican Party, then its successor, the Whig Party. The followers of William H. Crawford were also identified, though they never coalesced into any sort of larger organization and mostly existed in Georgia, Crawford's home state, though they found support among the
Friends of Reform:
In 1820, these were Republican candidates in Pennsylvania, mostly in Bucks County, opposed to the present administration.
New School / New School Republican / Old School / Old School Democrat / Old School Republican:
Used in Pennsylvania throughout the 1810's. They were often in opposition to the Constitutionalists. (See also: Crucible of American Democracy: The Struggle to Fuse Egalitarianism and Capitalism in Jeffersonian Pennsylvania. Andrew Shankman. University Press of Kansas. 2004.)
Used in several states over the course of over 20 years.
"Prior to the election of 1802 there had been minor divisions based largely upon personal jealousies and the quest for offices; and a vague dissatisfaction with the Governor had developed. A new cause of dissension became prominent in 1803 and 1804 as the legislature began to attempt modifications in the judicial system and to use its powers of impeachment against the judges of the State courts. McKean's opposition to most of these measures alienated many Republicans; and some of his supporters sought Federalist aid to redress the political balance." (The Keystone in the Democratic Arch: Pennsylvania Politics, 1800-1816. Sanford W. Higginbotham. 1952. p 49)
"The election of 1803 found the Republican splits becoming deeper and more widespread. The quarrel over Federal patronage in Philadelphia nearly reached the point of an open breach, while the Rising Sun movement against Leib gained added strength in Philadelphia County. In Lancaster some of the State officeholders made an unsuccessful attempt to organize a third party movement in support of McKean. The Federalists for the most part abandoned active politics, although the dissident Republican factions courted their aid." (Higginbotham, p 58)
Used in several states over the course of over 20 years.
In Rhode Island in 1807 and 1808 this was a splinter party formed by a combination of those republicans who were supporters of Governor James Fenner, combined with Federalists.
In New Jersey, for several years, from 1807 through 1822, this was a quasi-merged group between Federalists and Republicans, similar to the Quids in Pennsylvania.