Rather, former Whigs (like Abraham Lincoln) and those who had dallied with Free Soil (such as Salmon P. Chase and Charles Sumner) soon found a political home in the Republican Party. A gentlemen by the name of Alvan Earle Bovay (1818-1903) is credited with founding the party by calling for a meeting in Ripon, Wisconsin in early 1854, in opposition to the Kansas-Nebraska Act. That measure was then a bill (the "Nebraska" bill) pending before Congress that would allow a popular vote on whether the Kansas and Nebraska territories could have slavery, in contradiction to the Compromise of 1820.
"This meeting was held on Wednesday, March 1, 1854,: wrote A.F. Gilman of Ripon College in The Origin of the Republican Party in 1914. "What took place may be best expressed by the resolution that was adopted, which is as follows: 'Resolved, That of all the outrages hitherto perpetrated or attempted upon the North and freedom by the slave leaders and their natural allies, not one compares in bold and impudent audacity, treachery and meanness with this, the Nebraska Bill; as to the sum of all its villainies it adds the repudiation of a solemn compact held as sacred as the constitution itself for a period of thirty-four years.' "
Should the bill pass (which it did), the Ripon meeting further agreed that a new political party would be necessary to oppose it and the expansion of slavery. The meeting was not, of course, the only such meeting in the country, nor the only one to express those strong sentiments, but it's generally considered the first one out of the gate. Also, Bovay helped cement the name of the party with a timely letter to Horace Greeley, who endorsed the name in the June 24, 1854, issue of the highly influential Weekly Tribune.
The Republican Party has fielded a presidential candidate every election since 1856, claiming the prize 23 out of the last 39 elections.