We use down mostly as a preposition or adverb. It means ‘in or moving to a low or lower position or level’:
We ran down the hill like two little children.
Jamie, put that vase down! You’ll break it!
We can also use down to mean ‘towards the south’, ‘generally in the south’, or ‘towards the place where a river meets the sea’:
When you were living in Spain, did you ever go down to Granada?
We went on a cruise down the Nile for seven days. It was fantastic.
In informal situations, we can use down to talk about a quick trip to a destination which we consider to be less central than where we are. In this meaning, we can use it with or without to. Without to is less formal:
I’m just going down (to) the shop. Do you want anything?
Are you going down (to) the golf club tonight?
Down can also mean ‘along’. We can use it with from:
Mila’s office is just down the corridor, second door on the left.
They live just down the street from our house.
A good dictionary will tell you more meanings of down, especially when it is used as part of a phrasal verb.
Downwards is an adverb. It means ‘movement towards a lower position’:
The garden slopes gently downwards in the direction of the river.
Could you point the reading lamp downwards? It’s shining in my eyes.
In American English, downward (with no final -s) is used as an adverb:
Since the nineteen seventies, our country has really taken some steps downward.
In British English, downward (with no final -s) is normally only used as an adjective:
The statistics for violent crimes have shown a downward trend in the last two years. (The numbers have gone down.)