If you want a function that returns the browser as well as the version, here is an improvement from the original answer:
navigator.browserInfo = 
(
    function()
    {
        var browser = '';
        var version = '';
        var idString = '';
        var ua = navigator.userAgent;
        var tem = [];
        var M = ua.match(/(opera|chrome|safari|firefox|msie|trident(?=\/))\/?\s*(\d+)/i);
        //IE will be identified as 'Trident' and a different version number. The name must be corrected to 'Internet Explorer' and the correct version identified.
        //ie correction
        if(/trident/i.test(M[1]))
        {
            tem = /\brv[ :]+(\d+.?\d*)/g.exec(ua) || [];
            browser = 'Internet Explorer';
            version = tem[1];
        }
        //firefox
        else if(/firefox/i.test(M[1]))
        {
            tem = /\brv[ :]+(\d+.?\d*)/g.exec(ua) || [];
            browser = 'Firefox';
            version = tem[1];
        }
        //safari
        else if(/safari/i.test(M[1]))
        {
            tem = ua.match(/\bVersion\/(\d+.?\d*\s*\w+)/);
            browser = 'Safari';
            version = tem[1];
        }
        //If 'Chrome' is found, it may be another browser. 
        else if(M[1] === 'Chrome')
        {
            //opera
            var temOpr = ua.match(/\b(OPR)\/(\d+.?\d*.?\d*.?\d*)/);
            //edge
            var temEdge = ua.match(/\b(Edge)\/(\d+.?\d*)/);
            //chrome
            var temChrome = ua.match(/\b(Chrome)\/(\d+.?\d*.?\d*.?\d*)/);
            //a genuine 'Chrome' reading will result from ONLY temChrome not being null.
            var genuineChrome = temOpr == null && temEdge == null && temChrome != null;
            if(temOpr != null)
            {
                browser = temOpr[1].replace('OPR', 'Opera');
                version = temOpr[2];
            }
            if(temEdge != null)
            {
                browser = temEdge[1];
                version = temEdge[2];
            }
            if(genuineChrome)
            {
                browser = temChrome[1];
                version = temChrome[2];
            }
        }
        //There will be some odd balls, so if you wish to support those browsers, add functionality to display those browsers as well.
        if(browser == '' || version == '')
        {
            idString = 'We couldn\'t find your browser, but you can still use the site';
        }
        else
        {
            idString = browser + ' version ' + version;
        }
        alert('Your browser is ' + idString);
        //store the type of browser locally
        if(typeof(Storage) !== "undefined")
        {
            //Store
            localStorage.setItem('browser', browser);
            localStorage.setItem('version', version);
        } 
        else
        {
            alert('local storage not available');
        }
    }
)();
With this, it also stores the result locally, so this check is not necessary to run every time.