Abraham Lincoln
The Democratic split was so severe during the 1860 elections that the party actually put two candidates up to run against each other in the presidential election. The candidates' names were Stephen Douglas and John Breckenridge.
The two candidates that are running for senator in Kansas are Greg Orman ans Dr. Milton Wolf. The election is in a few weeks away.
The running of two Democrats effectively split the party in the 1860 election. The two candidates received votes from the Democratic Party, while the entire Republican Party, which held the majority of the larger North, threw its weight behind Lincoln. Southerners feared his election would ruin their way of life, and seceded from the union.
If you are asking about current day (2016), the remaining two Democratic primary candidates are former First Lady and former Secretary of State Hillary Rodham Clinton, and current US Senator from Vermont Bernie Sanders.
Maine and Nebraska
Democratic and Republican
Yes. He was the Democratic nominee in 1860. (He also campaigned unsuccessfully for the Democratic nomination in the two elections before that.)
Mitt Romney and Barack Obama
Just to clarify, there are more than two people running for president, a fairly comprehensive list of candidates can be found on Wikipedia. John McCain (Republican candidate) Barack Obama (Democratic candidate)
When the only candidates who have a reasonable chance of winning an election are one of two parties, this is generally referred to as a two party system. The US is an example of a two party system in which the only candidates who stand a reasonable chance of winning are Republican or Democratic.
Eugene McCarthy & Robert Kennedy