Getting a browser's name client-side

JavaScript side - you can get browser name like these ways...

if(window.navigator.appName == "") OR if(window.navigator.userAgent == "")

This is pure JavaScript solution. Which I was required.
I tried on different browsers. It is working fine. Hope it helps.

How do I detect the browser name ?

You can use the navigator.appName and navigator.userAgent properties. The userAgent property is more reliable than appName because, for example, Firefox (and some other browsers) may return the string "Netscape" as the value of navigator.appName for compatibility with Netscape Navigator.

Note, however, that navigator.userAgent may be spoofed, too – that is, clients may substitute virtually any string for their userAgent. Therefore, whatever we deduce from either appName or userAgent should be taken with a grain of salt.

var nVer = navigator.appVersion;
var nAgt = navigator.userAgent;
var browserName  = navigator.appName;
var fullVersion  = ''+parseFloat(navigator.appVersion); 
var majorVersion = parseInt(navigator.appVersion,10);
var nameOffset,verOffset,ix;

// In Opera, the true version is after "Opera" or after "Version"
if ((verOffset=nAgt.indexOf("Opera"))!=-1) {
   browserName = "Opera";
   fullVersion = nAgt.substring(verOffset+6);
   if ((verOffset=nAgt.indexOf("Version"))!=-1) 
     fullVersion = nAgt.substring(verOffset+8);
}
// In MSIE, the true version is after "MSIE" in userAgent
else if ((verOffset=nAgt.indexOf("MSIE"))!=-1) {
   browserName = "Microsoft Internet Explorer";
   fullVersion = nAgt.substring(verOffset+5);
}
// In Chrome, the true version is after "Chrome" 
else if ((verOffset=nAgt.indexOf("Chrome"))!=-1) {
   browserName = "Chrome";
   fullVersion = nAgt.substring(verOffset+7);
}
// In Safari, the true version is after "Safari" or after "Version" 
else if ((verOffset=nAgt.indexOf("Safari"))!=-1) {
   browserName = "Safari";
   fullVersion = nAgt.substring(verOffset+7);
   if ((verOffset=nAgt.indexOf("Version"))!=-1) 
     fullVersion = nAgt.substring(verOffset+8);
}
// In Firefox, the true version is after "Firefox" 
else if ((verOffset=nAgt.indexOf("Firefox"))!=-1) {
    browserName = "Firefox";
    fullVersion = nAgt.substring(verOffset+8);
}
// In most other browsers, "name/version" is at the end of userAgent 
else if ( (nameOffset=nAgt.lastIndexOf(' ')+1) < (verOffset=nAgt.lastIndexOf('/')) ) {
    browserName = nAgt.substring(nameOffset,verOffset);
    fullVersion = nAgt.substring(verOffset+1);
    if (browserName.toLowerCase()==browserName.toUpperCase()) {
       browserName = navigator.appName;
    }
}
// trim the fullVersion string at semicolon/space if present
if ((ix=fullVersion.indexOf(";"))!=-1)
    fullVersion=fullVersion.substring(0,ix);
if ((ix=fullVersion.indexOf(" "))!=-1)
    fullVersion=fullVersion.substring(0,ix);

majorVersion = parseInt(''+fullVersion,10);
if (isNaN(majorVersion)) {
    fullVersion  = ''+parseFloat(navigator.appVersion); 
    majorVersion = parseInt(navigator.appVersion,10);
}

document.write(''
                +'Browser name  = '+browserName+'<br>'
                +'Full version  = '+fullVersion+'<br>'
                +'Major version = '+majorVersion+'<br>'
                +'navigator.appName = '+navigator.appName+'<br>'
                +'navigator.userAgent = '+navigator.userAgent+'<br>');

From the source javascripter.net

EDIT: Since the answer is not valid with newer versions of jquery As jQuery.browser is deprecated in ver 1.9, So Use Jquery Migrate Plugin for that matter.

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Original Answer

jQuery.browser

jQuery.browser and jQuery.browser.version

is your way to go...