When I say to you ‘put down the book on the table’. You are asked to place it on the table. This is the literal meaning.
Put down the cup.
He put down the bundle on the floor.

It has several figurative meanings. I’ll mention a few.
‘To put down somebody’ is to silence or suppress her/him.
The Government has put down the Opposition successfully.
It is not easy to put down drinking and gambling.
He can’t be put down easily.
Stern measures were taken to put down the disturbances.

‘To put down’ also means ‘to record, write down’.
Let me put down your telephone number. I may forget it later.
Let’s put down what we have discussed.

When someone comes to you to collect some money for some cause and if you are willing to give, say Rs. 20, you may say ‘Please put me down for Rs. 20’. It means ‘Write my name and against it write Rs. 20’.
You may put me down for two tickets.
You may put me down for two plots.
Shall I put you down for the class on Tuesday?

‘To put your foot down’ is to assert your authority. When the clerk raised his voice in protest the boss put his foot down. When you put your foot down, you take a firm stand.
‘Put down to’ means to state that something is caused by something else.
I put down the loss to mismanagement.
I put down his unhappiness to domestic disharmony.
I put down his failure to lack of application.