This is Kate's jacket. = This jacket belongs to Kate.
That's our neighbors' dog. = That dog belongs to our neighbors.
Belong with means that things/people are similar and should be together. If one person in a romantic relationship says "You belong with me," it is like saying that it is destiny for the two people to be together.
You can also use belong with or belong in for putting an object into a category with other, similar objects:
That book about humans traveling to Mars doesn't belong in the history section. It belongs with the other science fiction books.
An object belongs IN a category, and belongs WITH other similar things.
While apple juice generally refers to the filtered, pasteurised product of apple pressing, an unfiltered and sometimes unpasteurised product commonly known as apple cider in the United States and parts of Canada may be packaged and sold as apple juice. In the U.S., the opposite is often seen; filtered and clarified juice (including carbonated varieties) may be sold as apple cider, thus there is an unclear distinction between filtered apple juice and natural apple cider.