Not sure how to edit your theme? Read this article first:
http://support.limoniapps.com/setting-up-dynamic-pricing/editing-your-theme

Find your cart template

Find your cart template in the Templates, Sections or Snippets folder. You're looking for the file that contains the markup that displays the cart page.
Typically it's called "cart-template.liquid" or "cart.liquid". 

Root

Find the root of the cart, which contains all cart lines and the subtotal. Add the class limoniapps-discountninja-cart-root to it.

Where you put this will be different for each theme. Here's an example from Debut:

Another example:

Line

A cart typically has a {% for %} loop that contains the markup for each cart line.
Find that for loop. It will be different for each theme, but typically looks like this:

On the line below the start of the for loop, the root element of the cart line will be defined. Discount Ninja expects that html element to have the class limoniapps-discountninja-cart-item. The content of this line will be different for each theme, but here's an example:

Variant

Below the line we put the limoniapps-discountninja-cart-item class on, we need to add a new line. This line is as follows:

<span class="limoniapps-discountninja-cart-item-variant"style="display:none;">{{item.variant.id}}</span>

Note 1: If your cart is structured as a table (i.e. it has rows <tr> and columns <td>) then you should add the above html inside the first <td> element.
Note 2:
we're assuming here that the variable "item" is available. It is declared in the "for loop" we discussed before. Double check that we're adding this line inside that "for loop", which starts with {% for item in cart.items %} and ends with {% endfor %}.

Product price

Note that the product price on a cart is different from the line price. The line price is the price of the product times its quantity.
Some themes show the product price on the cart, some do not. Look for a section that contains {{ item.price }}. The exact html varies per theme.
If your theme includes such a section, add a class limoniapps-discountninja-cart-item-product-price to the html element of that section.

Product price comment

You can optionally add a section where the product comment will be displayed. This comment can be defined per promotion and allows you to show on a cart which promotion is applied to which product (example: 20% off thanks to our winter promotion!). Discount Ninja looks for a section with the class limoniapps-discountninja-cart-item-product-comment and adds the comment at that location.
The markup for that section is as follows:

<div class="limoniapps-discountninja-cart-item-product-comment"></div>

Line price

Note that the product price on a cart is different from the line price. The line price is the price of the product times its quantity.
Some themes show the line price on the cart, some do not. Look for a section that contains {{ item.line_price }}. The exact html varies per theme.
If your theme includes such a section, add a class limoniapps-discountninja-cart-item-line-price to the html element of that section.

Line price comment

You can optionally add a section where the line comment will be displayed. The markup is as follows:

<span class="limoniapps-discountninja-cart-item-line-comment"></span>

Subtotal

Find the section that contains the subtotal. This section will include a reference to {{ cart.total_price }}. The exact html varies per theme.
The subtotal section needs a class limoniapps-discountninja-cart-subtotal-price.

Subtotal comment

You can optionally add a section where the subtotal comment will be displayed. This comment can be defined in the settings of the app (example: Discount applied. 30% off!).
Discount Ninja looks for a section with the class limoniapps-discountninja-cart-item-subtotal-comment and adds the comment at that location.

<span class="limoniapps-discountninja-cart-item-subtotal-comment"></span>

End result

The end result should look like this:

<div class="... limoniapps-discountninja-cart-root">

...

{% for item in cart.items %}

<div class="... limoniapps-discountninja-cart-item">
     <span class="limoniapps-discountninja-cart-item-variant"style="display:none;">{{item.variant.id}}</span>

...

<div class="... limoniapps-discountninja-cart-item-product-price">
... {{item.price | money }} ...
</div>
<span class="limoniapps-discountninja-cart-item-product-comment"></span>

<div class="... limoniapps-discountninja-cart-item-line-price">
... {{item.line_price | money }} ...
</div>
<span class="limoniapps-discountninja-cart-item-line-comment"></span>

{% endfor %}

<div class="... limoniapps-discountninja-cart-subtotal-price">
... {{cart.total_price | money }} ...
</div>
<span class="limoniapps-discountninja-cart-subtotal-comment"></span>

</div>

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