How to connect to a remote computer via ssh without typing password

Hello,

I am a newbie on linux and I would like to do the following thing :

Computer A
Computer B

I run a script on the computer A to make backup of data on computer B.
I would like to create private key and public key with ssh in order to not
type the password and put the task in a cron.

But I do not understand how to do it as well as the risk of doing it .
Because suppose someone not friendly
manage to take control of A , then he will have a full access on computer B.

Thank you for your comprehension and help.

Best Regards
T