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.
This is a game for teams of two, regardless of age.Each team stands side by side, and has their inside hands tied together.The person on the left has his right hand tied to the left hand of the person next to him. Each team is given a box, wrapping paper, ribbon, scissors and tape. With only one hand each, they have to completely wrap the package and tie it with a bow. The team that wraps the package correctly and quickest wins.