Look up diskless booting, it's a very general thing on linux (i.e. not Pi-specific, although there are plenty of Pi-specific tutorials on the 'net.

As a minimal first step, install an NFS server (which can be X86, Pi, other) on your LAN and make sure you can mount it from the pi ("mount $SERVER:/some/dir /mnt/tmp"). Then copy the contents of a Pi SD card to the server, make it exportable (see '/etc/exports') then edit '/etc/fstab' on the Pi to mount the (now remote) copy of the SD card instead of the usual root. That should get you started - beyond that, with some of the Pi's you don't even need to have an SD card installed (however you'll then need to set up things like DHCP and TFTP on your server).