Can someone please explain?

Does the Firefox browser share google chrome applications and tools? Whenever I connect to websites, unrelated to google or its browser, google chrome, I notice, paying attention to the bottom-left corner of the Firefox browser, that my browsing sessions have something to do with google and its browser. On another session, I noticed, that my browsing sessions had something to do with twitter.com, but I don't tweet. I don't have an account with Twitter. I'm not interested. Does the problem have to do with plug-ins? If so, should I disable all plug-ins. My understanding is that plug-ins allow for better website/user interface. Which plug-ins are secure? Can someone please explain?