You can use a lightbox to do it.

There is a simple example in this link:

http://lokeshdhakar.com/projects/lightbox2/

Seems that you are missing some CSS rules.

Let's have a look on this line:

popup.classList.toggle("show");

The toggle function here will add a class show class if the element popup does not have one, otherwise remove it.

That means the item with class popuptext should not display by default and unless the show class is added.

Therefore, you should define 2 CSS styles like the following:

.popuptext.show {
  display: inline-block; // show element when you have the class name 'show'
}
.popuptext {
  display: none; // hide by default
}

Here is a working example: https://jsfiddle.net/3bmqdcsg/

You can use a fancybox to do it.

a {
  color: #FFFFFF;
  text-decoration: none;
}

.primary-btn {
  background-image: -moz-linear-gradient(0deg, #235ee7 0%, #4ae7fa 100%);
  background-image: -webkit-linear-gradient(0deg, #235ee7 0%, #4ae7fa 100%);
  background-image: -ms-linear-gradient(0deg, #235ee7 0%, #4ae7fa 100%);
}

.primary-btn {
  line-height: 36px;
  padding-left: 30px;
  padding-right: 30px;
  border-radius: 25px;
  border: none;
  color: #fff;
  display: inline-block;
  font-weight: 500;
  position: relative;
  -webkit-transition: all 0.3s ease 0s;
  -moz-transition: all 0.3s ease 0s;
  -o-transition: all 0.3s ease 0s;
  transition: all 0.3s ease 0s;
  cursor: pointer;
  text-transform: uppercase;
  position: relative;
}

.primary-btn {
  color: #fff;
  border: 1px solid #fff;
  -webkit-transition: all 0.3s ease 0s;
  -moz-transition: all 0.3s ease 0s;
  -o-transition: all 0.3s ease 0s;
  transition: all 0.3s ease 0s;
}

.primary-btn:hover {
  background: transparent;
  color: #235ee7;
  border-color: #235ee7;
}

<link href="https://cdnjs.cloudflare.com/ajax/libs/fancybox/3.2.0/jquery.fancybox.min.css" rel="stylesheet" />
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/fancybox/3.2.0/jquery.fancybox.min.js"></script>

<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/fancybox/3.2.0/jquery.fancybox.min.css">
<a data-fancybox="gallery" class="primary-btn" href="//source.unsplash.com/user/erondu/300x300">Demo</a>

Instead of the onclick attribute, you could assign a class popup_image to your img and attach a click handler when the DOM is ready.

I took the liberty to remove scripts and tags that were not neccessary to demonstrate the result.

$(document).ready(function() {

  $(".popup_image").on('click', function() {
    w2popup.open({
      title: 'Image',
      body: '<div class="w2ui-centered"><img src="' + $(this).attr('src') + '"></img></div>'
    });
  });

});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.3/jquery.min.js"></script>
<script type="text/javascript" src="http://w2ui.com/src/w2ui-1.4.2.min.js"></script>
<link rel="stylesheet" type="text/css" href="http://w2ui.com/src/w2ui-1.4.2.min.css" />

<img class="btn popup_image" style="width:100px; height:100px; border-radius:4px;" src="http://lorempixel.com/g/200/200/"></img>

The span elements should be completely removed and its classes placed on the image elements themselves.

Also, you have a nested section element that isn't doing anything for you.

Lastly, do not use HTML heading elements (<h1>...<h6>) because of the way they style the text. Formatting is the job of CSS. Instead of headings, it is more appropriate to surround each image and its accompanying text with figure and figcaption elements.

img { 
  width:200px;
  border:1px solid black; /* This is only added for testing purposes*/
}

.thumbnail:hover {
   width: 500px;
   height:auto;
   position:relative;
   /* push image to the right by 1/2 the screen width and 1/2 the image width */
   margin-left:calc(50% - 250px);
}

<section id="main">
  <div class="inner">
    <div class="box alt">
      <div class="row 50% uniform">
         <div class="4u">
           <figure>
             <img src="https://pbs.twimg.com/profile_images/562466745340817408/_nIu8KHX.jpeg" alt="" class="thumbnail">
             <figcaption>Marble</figcaption>
           </figure>
         </div>

         <div class="4u">
           <figure>
             <img src="http://www.critterbabies.com/wp-content/gallery/kittens/cats-animals-kittens-background-us.jpg" alt="" class="thumbnail">
             <figcaption>Marble</figcaption>
           </figure>
         </div>
         
         <div class="4u">
           <figure>
             <img src="http://www.warrenphotographic.co.uk/photography/bigs/08482-Fluffy-ginger-female-kitten.jpg" alt="" class="thumbnail">
             <figcaption>Marble</figcaption>
           </figure>
         </div>
         
       </div>
     </div>
   </div>
 </section>

What you are looking for is called a lightbox. Their are many good tutorials on how to make a pure css one, here is a few of them: http://andornagy.com/pure-css-image-lightbox/
http://www.designcouch.com/home/why/2013/11/01/responsive-css3-lightbox-with-no-javascript/
http://www.thecssninja.com/xhtml/futurebox
The problem with using :target as a CSS click event is that it has some downsides such as page jumps or browser history.
You can avoid the downsides of :target by using the checkbox hack:
Make a checkbox and hide it:

<input type="checkbox" id="check" style="display:none;">

Then, make the image you want to have a lightbox for, and wrap it in a <label>

<label for="check">
<img src="http://lorempixel.com/400/400/" width="200">
</label>

Now, write the HTML for the lightbox:

<label for="check">
<div id="cover">
<div id="box">
<img src="http://lorempixel.com/400/400/" width="400">
</div>
</div>
</label>

And now, for the CSS magic! Create the lightbox css:

#cover{
position:absolute;
top:0;
bottom:0;
left:0;
right:0;
background:rgba(0,0,0,0.5);
display:none;
}
#box{
position:absolute;
top:0;
bottom:0;
left:0;
right:0;
margin:auto;
width:400px;
height:400px;
border:10px solid white;
}

This creates and centers the lightbox.
Now you need to add the click event:

#check:checked ~ label #cover{
display:block;
}

This CSS means, If #check is checked (:checked selector), find the sibling (~) with a id of #cover inside a label element and apply the rule to it. That's it!
Your coding should look like this:

<input type="checkbox" id="check" style="display:none;">
<label for="check">
    <img src="http://lorempixel.com/400/400/" width="200">
    </label>
<label for="check">
    <div id="cover">
    <div id="box">
    <img src="http://lorempixel.com/400/400/" width="400">
    </div>
    </div>
    </label>

And CSS:

  #check:checked ~ label #cover{
    display:block;
    }
#cover{
    position:absolute;
    top:0;
    bottom:0;
    left:0;
    right:0;
    background:rgba(0,0,0,0.5);
    display:none;
    }
    #box{
    position:absolute;
    top:0;
    bottom:0;
    left:0;
    right:0;
    margin:auto;
    width:400px;
    height:400px;
    border:10px solid white;
    }

SEE THIS JSFIDDLE