Assuming you are inside the network that you are asking about, you can go to a website like whatismyip.com and see what your public IP (Internet Protocol) address is. Unless you have requested and are paying for a static IP (one that doesn't change) you likely have a dynamic IP address (one that changes at given points over time). The IP address is like your street address. Information that leaves your network has a return (IP) address on it.
On the opposite side of this public IP address is your internal network. If it is using class A (10.0.0.0), class B (172.16.0.0 - 172.31.255.255), or class C (192.168.0.0 - 192.168.255.255) private networks, those IP addresses do not travel out into the Internet. Your router uses NAT (Network address translation) to convert between private and public network addresses.
So the short answer is no.
However, by looking at the information a web browser advertises, you can get an idea of the computer type, OS, and browser that is being used. If everyone in a network connects to your server, and you collect this information, you have an idea of whats there. A way to hide this is to use a proxy.
In order for me to get inside the network and see what's there I could use a reverse tunnel, then browse the network from whatever computer I reverse tunneled to inside your network. Realize this is an active attack on your network, not just passive information gathering. Defenses against this attack typically include software patches/updates, firewalls, and IPS/IDS. Let me know if you have any further questions.