Adding a Google Plus badge to your Google Site can be a challenge. There are a few clever hacks out there for getting it to work, but in the interest of keeping everything up to date, it's best to use the guidelines when embedding a badge.
This gadget will create a badge that allows your website's viewers to interact with you on Google Plus. There are some parameters to enter, but they key ones are your profile's URL, and the width you want. You can use the gadget by using the Insert - Gadgets - More Gadgets... menu option, and choosing "Add a gadget by URL." Enter this URL into the window:
For those who are interested, here's the source code:
Apps Script >