Network Stuff is a free, portable, open-source toolkit which comes packed with useful network-related functionality.
There are graphical Ping and TraceRoute tools, for instance. It only takes a couple of clicks to display all the open network and internet connections on your system. And if you don't recognise a particular remote IP address, then the DNS tab will try to resolve it to a domain or machine name, while the WhoIs tab provides the appropriate registration records for whatever domain you specify.
If you just want to monitor how your network is behaving over time then the Stats tab may help, as it displays TCP, IP, UDP and ICMP real-time stats, while also charting activity on whatever network interface you choose.
And exploring the various tabs and buttons just keeps revealing more functions: TCP and UDP telnet, TCP/ UDP/ ICMP/ CGI network scanning, packet capture, Wake On Lan, IP address discovery, packet forgery, interactive TCP/ UDP (view traffic and selectively allow or block whatever packets you like), and more.
Network Stuff's interface, unfortunately, isn't quite good enough to cope with all of this. There are no menus, no toolbars, so discovering the features you need requires much exploring of the many tabs and buttons. And these aren't always intuitively named, which means the initial learning curve is steeper than we'd like.
This shouldn't put you off using the program, though. Network Stuff deserves a place on most PCs just for its simpler functions (DNS, Whois, IP discovery and so on), which are all very easy to use; the rest you can learn when you need it, or if you just feel like exploring.