In Javascript, how do I determine if my current browser is Firefox on a computer vs everything else?

What you're after is known as browser detection:

if ($.browser.mozilla) { ... 

However, browser sniffing is discouraged, as its easy to spoof the user agent, i.e. pretend to be another browser!

You'd best use feature detection, either in your own way, or through the jQuery.support interface: http://api.jquery.com/jQuery.support/

Here's an article on extending it for your own use: http://www.waytoocrowded.com/2009/03/14/jquery-supportminheight/

Edit:

Found this post as well which helps: When IE8 is not IE8 what is $.browser.version?

I am doing some thing like below;

function checkBrowser(){
    let browser = "";
    let c = navigator.userAgent.search("Chrome");
    let f = navigator.userAgent.search("Firefox");
    let m8 = navigator.userAgent.search("MSIE 8.0");
    let m9 = navigator.userAgent.search("MSIE 9.0");
    if (c > -1) {
        browser = "Chrome";
    } else if (f > -1) {
        browser = "Firefox";
    } else if (m9 > -1) {
        browser ="MSIE 9.0";
    } else if (m8 > -1) {
        browser ="MSIE 8.0";
    }
    return browser;
}

Like this: Check for Firefox. Or some other browser.

 window.onload = function() {
          //  alert(navigator.userAgent);
            if (navigator.userAgent.indexOf("Firefox") > 0) {
                alert("ff");
            }
        }