According to Samsung's head of research and development for software and services Dr. Injong Rhee, Bixby is an "intelligent user interface" and is not designed to compete with Siri, Cortana, or other digital assistants. While those assistants are dedicated to helping you seek out information, Bixby's purpose is to make your Samsung phone easier to use. Bixby knows what's on your screen and what to do with it. The goal, according to Rhee, is to allow the user to use their voice to do anything that they could do through touch.
Samsung doesn't want Bixby limited to phones, either. The company would like to eventually take the AI to appliances, TV remotes, and other gadgets. Like with your phone, Bixby could help you use the functions in the devices around your house that you never knew were there or weren't sure how to use. Rhee envisions a future where all of your devices are connected and you can ask your washing machine to ask your phone to make a call. And all of this would be accessed through a dedicated Bixby Button.
Until we're given a demonstration, it's difficult to understand exactly what Bixby can do, but Rhee gives a pretty good idea. He calls Siri, Cortana and the like "glorified extensions of search," and while that isn't exactly what those are, it tells us what Bixby isn't.