There are different ways to launch the browser.
First of all , are you using "Keyword Driven Testing" or writing your own script ?
If you are using "Record and Play" then you don't have to bother about it , Testcomplete will take care of this.
If you are writing script then you have to write a code to launch it explicitly.
You can add Browser in "Tested Application" and use Run() function to launch it.
There are some very good videos posted on TestCompletes webpage.
Here is a link for the same:
http://support.smartbear.com/screencasts/testcomplete/Also , You can script like this [I don't have TestComplete installed on my machine , otherwise I could have written the complete function... but here is the small snippet.
Sub Test
Dim page, URL
URL = "
www.smartbear.com" Browsers.Item(btIExplorer).Run(URL)
Set page = Sys.Browser("*").Page("*")
End Sub
I would highly recommend you to go through the help files provided by testcomplete.