Horizontal relationships are the material ones we have with other people.
Vertical relationships are the ones we have with other spiritual beings who are not in this material world.
Note: Since we are all spiritual beings, all material relationships have a Spiritual component depending on the extent that the two people "emotionally connect"
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This could be either literal or figurative.
Literally, they are directions that are perpendicular (offset by an angle measuring 90 degrees) to each other. On a map, vertical would be north-south and horizontal would be east-west. To a standing person, vertical would be up-down and horizontal would be left-right (or across or sideways). A useful way to remember the difference is that going vertical gives some people vertigo (a dizzy sense) when they get high up, and horizontal is like the horizon (the line where the ocean or earth meets the sky).
Figuratively speaking, a vertical relationship is one that's between you and God. A horizontal one is between you and other people.
Horizontal is the relational dimension of life in a human level the vertical can only be a spiritual relationship one with God, who is spirit.