Answer: Lincoln believed slavery should not spread to new states, while Douglas believed each state should decide the matter for itself.
History/context:
Senator Stephen Douglas proposed a bill in Congress in 1854 that became the Kansas-Nebraska Act. The Kansas-Nebraska Act granted popular sovereignty to the people in the Kansas and Nebraska territories as they would enter the Union as states, letting them decide whether they'd allow slavery. Douglas' support for the Kansas-Nebraska Act was indicative of his approach on the issue of slavery.
In a speech Lincoln gave during the 1858 campaign for the US Senate, challenging Illinois' incumbent Senator Stephen Douglas, Lincoln had said of slavery, "Although volume upon volume is written to prove slavery a very good thing, we never hear of the man who wishes to take the good of it, by being a slave himself." Lincoln was morally opposed to slavery, but he also recognized that slavery was permitted by the existing law of the land, the US Constitution. So Lincoln's initial position on slavery was to stop the spread of it.
Lincoln did not manage to unseat Douglas from his seat in the Senate in 1858. However, Lincoln did succeed in winning election as President in 1860. When Lincoln was elected, states in the South moved to secede from the Union, which brought about the Civil War.
The progress of the Civil War made Lincoln increasingly strong in his stance against slavery. The war initially was about preserving the Union, but later, with Lincoln's Emancipation Proclamation (1863), was declared also to be about ending slavery.
