Javascript 'reset' function not working properly

Because you reset it to 0 and don't increment it on the next update. The quick fix would be

 $('#reset').click(function() {
    currentIndex = 0;

    $("h1").removeClass().addClass(weights[currentIndex]);
    $("h1").html(weights[currentIndex]);
    currentIndex=1;
  });

The proper way would be to do the incrementing at the beginning of the $('#text').on('click', function(e) { function

--Edit for clarification request--
This is how I would personally recommend writing it:

$(function () {
    var weights = ["jHairline", "jThin", "jLight", "jMedium"];
    var currentIndex = -1;
    $('#text').on('click', function (e) {
        currentIndex=(currentIndex+1)%weights.length;
        $("h1").removeClass().addClass(weights[currentIndex]);
        $("h1").html(weights[currentIndex]);

        e.preventDefault();
    });

    $('#reset').click(function () {
        currentIndex = 0;

        $("h1").removeClass().addClass(weights[currentIndex]);
        $("h1").html(weights[currentIndex]);
    });
});

The 2 lines I added/modified were var currentIndex = -1; and currentIndex=(currentIndex+1)%weights.length; .

So essentially, you are incrementing the number by 1 when the text.click function starts. If you start it at -1 at the beginning, then when text.click enters, it will increment it to 0. And setting it to 0 in reset will make sure it increments to 1 next time text.click is ran.

Further,

 if (currentIndex == weights.length - 1)
           currentIndex = 0;
        else
            currentIndex++;

is a bit more concise and clear when written as currentIndex=(currentIndex+1)%weights.length; . It is adding 1 to the number, and then modding (taking the remainder when dividing) to loop it back around to 0 once weight.length is hit.