The Kansas-Nebraska Act declared that "popular sovereignty" would decide whether Kansas would be a slave or free state when admitted to the Union. And people on both sides of the issue flooded into the Kansas territory in order to weigh any potential vote in favor of their cause.
By 1855 there were actually two competing governments in Kansas, and things turned violent the following year when an armed force in favor of slavery burned the "free soil" town of Lawrence, Kansas.
The fanatical abolitionist John Brown and his followers retaliated, executing several pro-slavery men at Pottawatomie Creek, Kansas in May 1856.
The violence even spread into the US Capitol. In May 1856 a congressman from South Carolina violently attacked a Massachusetts senator with a cane in response to a fiery speech about slavery and the unrest in Kansas.
Violent outbreaks continued until 1858, and it is estimated that approximately 200 people were killed in what essentially a minor civil war (and a precursor to the American Civil War).
The term "Bleeding Kansas" was coined by the influential newspaper editor Horace Greeley, editor of the New York Tribune.