Go here
http://www.goteamspeak.com/index.php?page=downloadsAnd download Client depending what operating system you use. Then just install it what ever directory you want. Then wait it to be installed. Now, you need to know where to connect if you want to chat with someone. You may also make your own server. In that case download also TeamSpeakserver depending what operation system you use. Install it to what ever directory you want.
If you know where to connect, you don't need server then (when you have server, people connects to your computer) just select
Connections -> Quick Connect
Server address: insertaddresshere.org or IP 555.555.55.5 (examples only :p )
Nickname: MyNickname
Login: Nothing if you are not registered user (made by server admin)
Password: Nothing, if not server pass protected (same reason as above)
Connect.
And, well lol, make your mic is working before you even start to do this.
Your own server?
Easy:
Turn server on
Click icon with right button -> Your IP's -> Update IP's -> Check IP
-> Turn on TS client -> Connect to your IP
Tell pals to join your server.