Stuff built into Windows (XP)
ipconfig - The first place to check when you're having network issues. Use this to verify the IP address is set, and add the /all switch to make sure you have the correct DNS entries.
route - The second thing to look at when you're having network routing problems. With the print flag it shows you every identified route from your computer.
Process Explorer - This gives you an accurate and complete picture of how memory is allocated and everything a process has loaded. It will even tell you every Registry key a process touches, which can be tremendously handy.
[Side Note: Task Manager has numerous limitations, but two really stand out as truly egregious. First, it doesn't accurately report virtual memory. It actually tells you the private bytes a process has allocated, which is not the VM size. Second, it doesn't tell you external modules (DLL's, spawned applications) a process has loaded. If you see that an app is hanging you don't know if it's a problem in the application or one of the DLL's or applications it calls. Process Explorer plugs both these holes, and a whole lot more.]
Network Monitor - Allows you to capture and analyze TCP/IP packets.
Wireshark - A network protocol analyzer that takes the Network Monitor to new levels. It can be used with everything from TCP/IP to USB and FDDI. This is the ultimate tool, but it does have a steep learning curve.
iperf - This handy utility will measure the bandwidth, delay jitter and datagram loss between hosts. This is most useful in WAN scenarios so you can verify what the bandwidth is between locations, but you can also use it as an indicator of network congestion on a LAN. Note that this is also sometimes called lperf, with a lower case "L" at the beginning. The correct name starts with a lower case "i".
I also want to mention that a great collection of tools is available from Sysinternals. This was a private company making tools (such as Process Explorer and Network Monitor) to plug holes in the base offerings by Microsoft. Microsoft bought them in 2006, but the same people are still working on the tools so the quality is still great. There is a wide range of tools, so you'll probably find something useful. :-)