Depends on the setup. For a binary system, there's really only two setups. One with two stars close together, and the planet you're on orbiting the center of mass of the two stars. Tatooine from Star Wars is like this. So it would be mostly like Earth, just with two glowing orbs in the sky next to each other during the day instead of just one glowing orb.

The other configuration would be two stars further apart, and the planet orbiting one of them. For example if one of the gas giants in our solar system was heavy enough to start nuclear fusion. Such as what happened to Jupiter in the 2001 universe (Jupiter actually gets turned into a star in the sequel, 2010). Now, the outer star will revolve around the main star, but much slower than the inner planet revolves the main star. So like Jupiter it will rise and set at approximately the same time tomorrow as it does today. But at least as far as Earth and Jupiter goes, the outer star (Jupiter) will rise about 3-4 minutes earlier tomorrow, and then 3-4 minutes earlier the day after tomorrow, etc., which means over roughly a year it will drift from being in sync with the main star, to being completely out of sync with the main star, and everything in between in terms of outer star sunrise and outer star sunset. Since Jupiter takes about 12 years to go around the Sun, it will actually take about 13 months on Earth for the cycle to repeat.