Solution 1:

Downloaded TeamSpeak 3 file needs to be run - its an installer script.

1. Allow Execution of Installer

You have to change its permissions first, allowing it to be executed. You can do it in shell (chmod +x TeamSpeak3-Client-linux_amd64-3.0.16.run) or selecting Properties of the file, and allowing it like this:

Allow execution file as program

2. Run the Installer

Open your Terminal (you can press default shortcut of Ctrl+Alt+T), and go to directory where the file is located, eg:

cd Downloads

And run the installer, eg. like this:

./TeamSpeak3-Client-linux_amd64-3.0.16.run

3. Keep System Clean

TeamSpeak will be installed in current directory, and its probably a good idea to move it somewhere - /opt is good place to keep additional software like this (which didn`t come from "normal" Ubuntu packages):

sudo mv TeamSpeak3-Client-linux_amd64 /opt/

4. Run TeamSpeak

To run installed TeamSpeak enter:

/opt/TeamSpeak3-Client-linux_amd64/ts3client_runscript.sh

5. Create Launcher

You can permanently create a Launcher, for yourself:

gedit ~/.local/share/applications/TeamSpeak3.desktop

...or for all users on your system like this:

sudo gedit /usr/share/applications/TeamSpeak3.desktop

Put a content to this Launcher like this:

[Desktop Entry]
Name=TeamSpeak 3
Comment=TeamSpeak 3 VoIP Communicator
Exec=/opt/TeamSpeak3-Client-linux_amd64/ts3client_runscript.sh
Terminal=false
Type=Application
Categories=Network;Application;
Icon=/opt/TeamSpeak3-Client-linux_amd64/styles/default/logo-128x128.png

Remember to replace file and directory names accordingly to TeamSpeak version (here 3.0.16) and target architecture (here amd64).

(The default Icon is somewhat too pale - you can find another icon if you wish)

Solution 2:

Since I can't comment yet I'll post this as an answer. @madneon 's answer works but you need to put "" around the path in Exec otherwise it won't work (tested on Ubuntu 16.04) Seeing how this is one of the first results when searching for this 'problem'

To be clear:

[Desktop Entry]
Name=TeamSpeak 3
Comment=TeamSpeak 3 VoIP Communicator
--Exec="/opt/TeamSpeak3-Client-linux_amd64/ts3client_runscript.sh"--
Terminal=false
Type=Application
Categories=Network;Application;
Icon=/opt/TeamSpeak3-Client-linux_amd64/styles/default/logo-128x128.png