This would be mostly (*)nix based command set - Here we go...
Lets you check whether your internet connection is alive or lost
Example : ping techinfomedia.blogspot.com
Lets you trace packets and find if your connection is broken beyond your ip address. Also can be used to track the number of hops to get to the hostname
Example: traceroute techinfomedia.blogspot.com
Find out your ip address, your network interfaces, transferred and received data information
Example : ipconfig
Find out the dns name resolves to a IP - a way to test your dns server
Example: nslookup techinfomedia.blogspot.com
Does the same as above and provides other dns related information
Example: dig techinfomedia.blogspot.com
Find out all the open ports on your machine
Example: netstat -a
Display all the tcp based established connections on your machine
Example: netstat -nt
Telnet or connect to a machine at the specified port to find out whether that machine/server is working right
Example: telnet techinfomedia.blogspot.com 80
Ofcourse, there are many other options within each of these commands which you can find out through the manual pages by typing man {command}
Update: Some of the commands might be in /sbin or /usr/sbin paths and might not be in default PATH. So go and execute it from there! For Example: /sbin/ifconfig
