Basically, what the user (client) needs is to generate a public/private key pair. The server will know the public key, but only the client will know the private key.
When the client connects to the server, it tells its own public key. If this key is allowed (if it is between the know public keys list on the server), the server will send a randomic number to the client. This encripted number can only be decripted if the appropiate decription key is used, and this decription key is the client's private one. The client then uses then its own private key and decript the number. If this is done correctly, the server will grant the acces with no more questions. As you can see the system is safe, because the client never tells anybody about its private key; and this key cannot be inferred using the public one.

What must be done, then , is to generate a public/private key pair, and copy the public part into the appropiate place on the server side.
For doing this, on the user's home directory, on the client machine, type

local>ssh-keygen -t rsa -f .ssh/id_rsa

-t tells the type of encription
-f tells where to store the public/private key pairs. In this case, the .ssh directory on home is being used

A password will be asked; leave this part blank, just pressing
Now, go the .ssh directory, and you will find two new files: id_rsa and id_rsa.pub. The last one is the public part. Now, copy the public key to the server machine

local> cd .ssh
local> scp id_rsa.pub user@remote:~/.ssh/id_rsa.pub

Of course, this time you will need to enter the password.
Now, login into the server machine and go to the .ssh directory on the server side

local> ssh user@remote
remote> cd .ssh

Now, add the client's public key to the know public keys on the server

remote> cat id_rsa.pub >> authorized_keys

remote> chmod 640 authorized_keys

remote> rm id_rsa.pub

remote> exit

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