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.
The spacecraft used in the fourth manned mission in the United States Apollo space program, Apollo 10 holds the Guinness World Record for highest speed attained by any manned vehicle, at 24,791 mph (39,897 km/h), during the return from the Moon on May 26, 1969.