Where do i need to put my public rsa key on the server to allow passwordless ssh authentication?

I have been trying to get ssh working without a password on a Linux server. I have been looking at a lot of tutorials, but I have a lack of fundamental understanding of some key concepts.

I have an old rsa public key that I generated on my client server, which I am trying to use. I log in to the Linux server, and add my public rsa key to /home/user/.ssh/authorized_keys. I am still prompted for a password when attempting to log in.

My question is whether or not that is the correct authorized_keys file to add my key to. It seems odd that I would put it in my own user directory, but most explanations I have seen on setting up ssh tell me to put my public key in ~/.ssh/authorized_keys.

As a follow up question, is the public key that I generated years ago on my client my identifier for all shh connections or do I need to generate a new one?

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