Telnet is a tool that allows you to test connections to a variety of TCP based services on hosting and mailservers.
TCP stands for Transmission Control Protocol in the most simplest of terms means it's a reliable way of transferring data over the IP protocol. This type of connetion is used for the likes of web, ftp and email traffic and many others. Each of these services needs to be connected to via a port specific to that service, eg: web: 80 ftp: 21 email: 25, 143, 110 (not a complete list)
This is where telnet comes in as you use it to perform a basic test to check if your local system can at a minimum connect to the these services, if it can't you typically have a local issue of some sort preventing this connectivity, most of the time caused by firewall on the desktop system or router. There are other causes for example many ISP's will block port 25 forcing you to use port 587 to send email instead.
On to the telnet tests, this is reliant on having a "telnet client application" installed. Unfortunately the latest iterations of windows no longer install telnet as a default client, you can still add this to your windows client by using the "add/remove programs feature" as there is a built in client available.
Once installed you will need to open a command prompt, also known as a dos prompt, if on OSX you need to open the terminal application which will also provide you with a command line prompt.
Once presented with this prompt you type:
telnet host.domainame.com xx Where you enter your server name and the "tcp port" in the xx fields.
On our Australia based quadra servers, eg: telnet mail1.quadrahosting.com.au 25 (testing mail connection)
The response you will see should be: Trying 22.214.171.124... Connected to mail1.quadrahosting.com.au. Escape character is '^]'. 220 mail1.qnetau.com ESMTP
On windows you may also sometimes see a black screen, this typically means it has connected but ommitted the above response, if you cannnot connecd you will eventually get a timeout error.
You simply change the host and tcp port to suit what you want to test.
Another example: telnet nix4.quadrahosting.com.au 21 (here you are testing ftp)
Here the response would be: Trying 126.96.36.199... Connected to nix4.quadrahosting.com.au. Escape character is '^]'. 220 ProFTPD 1.3.4d Server (188.8.131.52) [::ffff:184.108.40.206]
You simply repeat the tests from here for every TCP port you need to test.