If I change my user-agent with a plugin, can sites still detect my browser?
By default, web browsers will send their standard user agent with each HTTP request they make.
However it's possible to use a browser extension (eg. User-Agent Switcher: Firefox or Chrome) or the in-built developer console to manually change the user agent that your browser will send with its requests.
Inspecting the user agent is the main method that websites use to determine which browser you're using, and so simply changing the user-agent your browser sends will typically be enough to trick most sites into thinking that you're using a different web browser than you really are.
However there are other, trickier (but perhaps more unreliable or less specific) ways of determining the browser you're on.
There are browser features which can be tested for, which some browsers have but others don't.
For example, if you're using Chrome and you changed your user agent to use the Internet Explorer 11 user agent, if a site simply checked the user agent, you would appear to be using IE 11.
Versions of the HTML5 canvas can reveal which browser/OS you're using as well. The plugins which you've got installed can help identify your system, and so on.
Sometimes your HTTP Headers can leak interesting information about your browser.
So in short, yes, just because you've changed your UA doesn't mean you can't be detected, however the vast majority of sites won't bother checking to that level.