There are a number of ways to start a Google Voice search. All of them require that you be on a Google search page. Since I'm always using Chrome, I usually just have to open a new tab which takes me to a page with a Google search bar on it.
From there, you can click on the microphone on the right side of the search bar and then speak your Google search.
I prefer just to use the keyboard shortcut in chrome to start a voice search: ctrl+shift+period. This automatically starts the search.
Lastly, Google has released an extension called Google Voice Search Hotword (Beta). It is supposed to allow you to just say "Ok, Google" like you do on your phone and open the voice search. However, I have not been able to get it to work on my computer, so I am unable to weigh in on how well it actually works.
Using your voice to control the computer is not yet for everyone, but it is starting to come along. Features like Google Voice Search are likely the beginning phases of this technology.