1. Download the client.
You can download the Teamspeak 3 Server client from Teamspeak 3's download page. Choose the client which suits your computer's OS and processor (Windows/Mac/Linux and 32-bit/64-bit).
2. Unzip the file.
Once you download the file it should be in a form of .zip-file. Unzip the file using your favorite tool. The contents of the unzipped file should look like this before you have launched the client. After you launch it the first time, it automatically creates more documents.
3. Launch the server
When you launch the server for the first time, you will be prompted by a window which includes three important lines of text. Login name (which is always serveradmin until changed), password (randombly generated gibberish) and Server Admin Token. I suggest you copy paste the fields to a .txt or similar document before closing the window.
The picture below shows that the program is running. You can distinguish the server client by a red dot it has instead of a green dot which is the mark of the user client.
4. Find out your IP address.
This is an easy step. Launch your favorite web browser, open google or similar search-engine and write "What is my IP?". The very first results should lead you to an web-page which contains your IP. Your IP is IPv4 and looks like XXX.XXX.XXX.XXX where "XXX" is a number between 0 and 255. An example would be 127.0.0.1 which is usually reserved for the slot 1 of Local Area Network (LAN).
5. Others connect to your server.
Give your IP address to those who wish to join your server. If you have a password on the server, give it to them as well. Open the Teamspeak 3 User client and press "Connections" on the top left corner of the program to open up a dropdown menu which includes a button "Connection". Press it to open the "Connect" menu.
In the Connect Menu write the IP in the Server Address field, choose your Nickname and write the password to Server Password field. You can click a button "more options" to show more options, but it is unnecessary. The additional options would include choosing your desired default channel (the channel you join automatically when joining the server) but that is unnecessary for most users.
Done.
Troubleshooting!
People can't join my server.
Check if your server is accessible to people. Begin with yourself. Open the Teamspeak 3 User Client and go to Connections > Connect and write to Server Address field "localhost" without "". If the server is running on your own computer you should be able to join the server. If you can't join the server, then it is not running. If you can join, then it is running and something is blocking the other people from joining your server.
Check that your Computer's Firewall doesn't block the Teamspeak 3 Server Client from communicating over the Internet. There are many different Firewall providers, so I won't list any help for