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.
Atal Bihari Vajpayee was born into a middle class family in Gwalior. His father was Krishna Bihari Vajpayee, a school teacher and poet, and his mother was Krishna Devi.He attended the Saraswati Shishu Mandir and later went to Victoria College from where he graduated with distinction in Hindi, English and Sanskrit.
He enrolled at DAV College, Kanpur to complete his post graduation and earned his M.A. in Political science. He was a patriot at heart and participated in Indias struggle for freedom even as a student and edited several nationalist Hindi newspapers.