below, under

Below and under can both mean lower than and most of the times either can be used.

In general, under indicates contact. Prefer under when something is covered or hidden by what is over it, and when things are touching.

• Sarah put the letter under her pillow.
• The toy car crackled under his feet.
• What color shirt are you wearing under your sweater?
• The whole village is under water.

With below there is usually a space between the two surfaces. Prefer below when one thing is not directly under another.

• They live below us. (we live on third floor and they live on the second.)
• The mirror was placed below the window.
• The path runs below a long brick wall.
• The sun had already sunk below the horizon.

Below and under can mean junior in rank. 

He is under me implies that I am his immediate superior. Below may or may not imply the same meaning. 

Use below in measurements of temperature and height, and in cases where one thinks of a vertical scale.

• The temperature is 2 degrees below zero.
• Fully natural places below sea level require a dry climate.
• The plane flew below the clouds.
• Tom's strength is below average for a child his size.

Use under to mean less than or younger than.

• There were under fifty people at the conference.
• You cannot watch this film if you are under 18.

Beneath can sometimes be used instead of under, but prefer using it for abstract meanings.

• Captain would think it beneath him to tell a lie. (unworthy of him)
• She married beneath her. (into a lower social class)
• Her behavior is beneath contempt. (really disgraceful)