under - below - beneath
Under is almost always a preposition. You use under to say that one thing is at a lower level than another, and that the other thing is directly above it. For example, you might say that an object on the floor is under a table or chair.
There's a cupboard under the stairs.
A path runs under the trees.
Underneath can be a preposition with a similar meaning to under.
We sat at a table underneath some olive trees.
Underneath can also be an adverb.
Let's pull up the carpet and see what's underneath.
Below can be a preposition. You normally use it to say that one thing is at a much lower level than another.
There's a tunnel 100 metres below the surface.
Below can also be an adverb.
They stood at the top of the mountain and looked at the valley below.
Beneath can be a preposition with a similar meaning to under or below. Beneath is more formal.
He could feel the soft ground beneath his feet.
Beneath can also be an adverb.
He stared out of the window at the courtyard beneath.