You will be able to share the HDD as a network drive but depending on the speed of your network you may not get very good performance from it (~13MB/sec on 100Mbit LAN). A better idea would be to get an external caddy. This is a case which you plug the HDD into (power and IDE cable) then connect to your main PC via USB2.0/Firewire. This will give you better performance (~35-40MB/sec) and you don't need to have it on all the time. This reduces power consumption and decreases the likelihood of losing the data (offline data can't be destroyed by a virus, for example). Here is an example caddy. If you do buy a caddy make sure it is compatible with IDE and 3.5" (rather than 2.5" laptop drive). Some caddies can handle both SATA and IDE drives but may require you to change internal connectors.
Your router may not support 1Gbit/s on the network ports - look into the manual or the technical specs of the router to find out if it does. If your router does not support 1Gbit/s on the network ports you'll need to either buy a router which does (may be expensive) or buy a gigabit switch (quite cheap for a bog standard 4 port switch). Then you connect each of the two PCs directly to the switch with good cable (Cat5e or Cat6) and connect the switch to the router with Cat5 cable.
"Network drive" in normal usage means a drive shared within a LAN (i.e. not one that's shared to the Internet). If you want to share it within the LAN that is easy - but sharing it so you can access it from the Internet it will be more difficult.
To share it on the local network (so you can access the files using the main PC) read this guide. You may need to change steps depending on the version of Windows installed on the PC.
To share on the Internet, you could set up an FTP server on the machine with the 500GB drive then log in to it with an FTP client (FileZilla provides software which can be used for the both the server and client). If you did this you would have to forward the FTP server port on your router so you could access it from the Internet. Note that this would be a potential security risk, as having ports open to the Internet can expose your system. Choose a port other than the default FTP port (21) and choose a strong password and read up about FTP server security.
You can do both at the same time (share the drive on the local network and on the Internet) at the same time - you do not need to make a choice of one or the other.
