How do I detect IE and Edge browser?

For me better this:

var uA = window.navigator.userAgent,
    isIE = /msie\s|trident\/|edge\//i.test(uA) && !!(document.uniqueID || document.documentMode || window.ActiveXObject || window.MSInputMethodContext),
    checkVersion = (isIE && +(/(edge\/|rv:|msie\s)([\d.]+)/i.exec(uA)[2])) || NaN;

Go run: http://jsfiddle.net/Webnewbie/apa1nvu8/

I use these functions, which work even if the user agent is set to something else.

if (document.documentMode) 
{
    console.log('Hello Microsoft IE User!');
}

if (!document.documentMode && window.msWriteProfilerMark) {
    console.log('Hello Microsoft Edge User!');
}

if (window.msWriteProfilerMark) 
{
    console.log('Hello Microsoft User!');
}

And this detects Chredge/Edgium (aka. Anaheim)

function isEdg()
{ 

    for (var i = 0, u="Microsoft", l =u.length; i < navigator.plugins.length; i++)
    {
        if (navigator.plugins[i].name != null && navigator.plugins[i].name.substr(0, l) === u)
            return true;
    }

    return false;
}

And this detects Chromium:

function isChromium()
{ 

    for (var i = 0, u="Chromium", l =u.length; i < navigator.plugins.length; i++)
    {
        if (navigator.plugins[i].name != null && navigator.plugins[i].name.substr(0, l) === u)
            return true;
    }

    return false;
}

I doubt you really need to detect the browser. But here it is anyway (don't really need to use a library):

// detect IE8 and above, and edge
if (document.documentMode || /Edge/.test(navigator.userAgent)) {
    alert('Hello Microsoft User!');
}