Putting vs. Resting
Russian uses different verbs for putting an object somewhere (an action) versus the object resting there (a state).
Putting something somewhere implies motion toward a destination and requires the Accusative case (answering where to?). An object already resting somewhere uses the Prepositional case (answering where?).
Hanging: вешать vs. висеть
To say you are hanging something up, use вешать + Accusative. If the object is already hanging, use висеть + Prepositional.
Standing: ставить vs. стоять
For objects that have a base and sit upright (like a vase or a tray), use ставить (or perfective поставить) + Accusative. Once placed, the object uses стоять (to stand) + Prepositional.
Lying flat: класть vs. лежать
For objects that lie flat or lack a base (like a blanket or book), use класть (to lay) + Accusative. When the object is resting, use лежать (to lie) + Prepositional.
The perfective verb: положить
The perfective partner of класть uses a different root: положить. It describes the completed action of laying something down. (Note: never use the root "лож" without a prefix!)
Placing objects inside and on top
Instead of prepositional phrases, you can use directional adverbs like наверх (up / on top) and внутрь (inside) to describe a destination.
From Action to State
These verbs often pair up when an action results in a resting state. The action takes the Accusative (or a directional adverb), and the resulting state takes the Prepositional (or a static location word like там).