These colorful political terms described subsets of the New York Democratic party in the 1830s and 1840s. The "Locofocos" originated in October 1835 at a rowdy Tammany Hall meeting. Tammany bosses tried to end the meeting by shutting off the lights, but a group of radicals responded by lighting "locofocos" (a type of match) and continuing on their own. They were fiercely opposed to monopolies (particularly in the banking world), and fought Tammany Hall on many local issues, despite their allegiance to the Democratic party.
The Locofocos (known more prosaically as the Equal Rights party) peaked between 1835 and 1837. They enjoyed little success at the ballot box, but their vituperative defense of democracy inspired many, particularly the incipient Northern laboring class, and Martin Van Buren incorporated some of their ideas into his monetary policy. Thereafter, Democrats were collectively nicknamed Locofocos. (Nathaniel Hawthorne called himself "the Locofoco Surveyor" in his preface to The Scarlet Letter).
The term "Barnburners" applied to several groups. It stemmed from an adage about a Dutch farmer who burned down his barn to rid it of rats, connoting a militant group obsessed with one issue. In the early 1840s, it designated New York Democrats opposed to state fiscal policy, but by the mid-1840s it described a more serious schism—Democrats against the expansion of slavery. These "Barnburners" would break off from the party to join the Free Soil campaign in 1848. Whitman was active in the campaign and supported Barnburner causes at the Brooklyn Daily Eagle, though the paper was owned by a "Hunker" (a traditional Democrat), which probably led to his dismissal in January 1848.
Bibliography
Byrdsall, Fitzwilliam. The History of the Loco-Foco or Equal Rights Party. New York: Clement and Packard, 1842.
Donovan, Herbert D.A. The Barnburners: A Study of the Internal Movements in the Political History of New York State and of the Resulting Changes in Political Affiliation, 1830–1852. New York: New York UP, 1925.
Trimble, William. "Diverging Tendencies in New York Democracy in the Period of the Locofocos." American Historical Review 24 (1919): 396–421.