Jquery random number

The "Play" button is good to start the ball rolling (I'm not certain if you were thinking of removing it entirely). To generate numbers periodically, use setInterval.

$(function(){
    var initialPeriod=500; // 0.5s
    var generateNumIval;
    function generateNum() {
        var numLow = $("#lownumber").val();
        var numHigh = $("#highnumber").val();

        var adjustedHigh = (parseFloat(numHigh) - parseFloat(numLow)) + 1;
        var numRand = Math.floor(Math.random()*adjustedHigh) + parseFloat(numLow);

        if (IsNumeric(numLow)
            && IsNumeric(numHigh)
            && (parseFloat(numLow) <=  parseFloat(numHigh)) 
            && (numLow != '') 
            && (numHigh != '')) 
        {
            $("#randomnumber").text(numRand);
        } else {
            $("#randomnumber").text("Careful now...");
        }
    }
    function run(period) {
        clearInterval(generateNumIval);
        generateNumIval = setInterval(generateNum, period);
    }

    $("#getit").click(function() {
        run(initialPeriod);
        return false;
    });
    ...

You can change the period (such as to increase the difficulty when the user clicks the correct number, or decreasing the difficulty when the user makes too many sequential mistakes) by calling run with a new period. If you want to change the period after generating each number, use setTimeout rather than setInterval.

To check a click on a number, register a click handler on #randomnumber that compares its val() to #MyNumber's val(). From there, take appropriate action as to whether it's a hit or miss. As Dan says, doing this for every click will create quite a bit of network traffic. Though only a small amount of data may be transmitted each time, the number of connections can cause a significant impact. Instead, have a "Stop" button and send the data if the user clicks it, or use an unload handler (one does not exclude the other).

Your server will crash and burn if you have more than a couple people playing this game. People can identify and click very fast (multiple times per second), but unless they live next to your server, you can't receive and respond to HTTP requests that fast, nor can your server handle hundreds or more per second from the multiple users.

Write the game in JavaScript and when they're done, send the totals (# of wrong clicks and # of right clicks, or whatever) to your server to save. Do your best to obfuscate how they're sent so that it's not trivial to make up scores.

There's a couple of things to look out for here. There's no reason why the random numbers can't be generated from the number the player has entered himself, or even better, a number generated by the game itself.

The way which you've done the placeholder text, using data and two event handlers is also somewhat messy. At a minimum you should be using .one to attach a one-time event handler for this, but it would be much better if you use the HTML5 placeholder attribute with a Javascript fallback.

Other than that, you're still missing significant amount of game logic in there. I won't advice you to work on this game for too long though - it's great as an exercise in working with JavaScript and jQuery, but otherwise not very worthwhile.

Oh, and just for fun, I also built my own version of this.

var hitCount = 0,
    missCount = 0;

function IsNumeric(n) {
    return !isNaN(n);
}

$("#getit").click(function() {
    var li = [],
        intervals = 0,
        n = parseInt($('#MyNumber').val());

    if (IsNumeric(n)) {
        setInterval(function() {
            li[intervals++ % li.length].text(Math.random() > .1 ? Math.floor(Math.random() * (10 + n) + (n / 2)) : n).attr('class', '');
        }, 500);
    }

    $('#randomnumber').empty();

    for (var i = 0; i < 5; i++) {
        li.push($('<li />').click(function() {
            var $this = $(this);

            if (!$this.hasClass('clicked')) {
                if (parseInt($this.text(), 10) === n) {
                    $this.addClass('correct');
                    $('#hitcount').text(++hitCount);
                } else {
                    $this.addClass('wrong');
                    $('#misscount').text(++missCount);
                }
            }

            $this.addClass('clicked');
        }).appendTo('#randomnumber'));
    }

    return false;
});

Crude yes, but it sort of works. Have a look at it here: http://jsfiddle.net/DHPQT/