The launcher is in a fixed position in the bottom right quadrant of the screen. Using the chat designer UI, you can move it anywhere within that quadrant.
The Launcher can also be moved to a different part of the page, like the bottom left, for example. This position can be altered via CSS. This is something your web designer should be able to put in place if you wish.
The CSS properties here shown can be changed to move the launcher. For example right can be changed to left: