Copying files while SSHed onto a computer

I am trying to upload some website files to a computer I am SSHed into and I'm not sure the correct command to use. I've tried looking up the cp command instructions but I can't work out what to put down my local machine as, since I of the understanding that my local IP address will be masked from the outside.

So, how would I go about finding out what my local IP address is for the purposes of copying files from my computer to a remote server via command-line access on the remote server? (I am assuming at this stage that the cp Linux command would be the best way to do so?)

Currently the only other way I have to upload files is via a C-Panel interface which is quite cumbersome and only allows the copying of one file at a time, which, as the website grows, is likely to become more of a significant problem.

Any help would be very, VERY much appreciated.