How to use page specific template

I want to vary the home page of my site from all others. Nothing major just a small design change. The codex says:

WordPress can be configured to use different Page Templates for different Pages. Toward the bottom of the Write > Page administration panel (or on the sidebar, depending on which version of WordPress you are using) is a drop-down labeled “Page Template.” From there you can select which Template will be used when displaying this particular Page.

What is write > page referring to? I assumed it was page edit but there is nothing in my editor to allow me to select a page specific template. Is template the same as theme?

Please can someone help me out. Thanks