em

Take this HTML as an example:

<div id="outer">
    <h1 class="heading">Outer heading</h1>

    <div id="inner">
        <h1 class="heading">Inner heading</h1>
    </div>
</div>

And the following CSS:

#outer { font-size: 30px; }
#inner { font-size: 20px; }
.heading { font-size: 3em; }

This is the result:

If you inspect the headings in dev tools, switch to computed tab, the outer heading has font size 90px and the inner has 60.

That's because em based on its parent's font size.
If I didn't specify a font size for #inner, the two heading would have the same size.

rem

rem is a little bit different. Let's change the .heading font size unit from em to rem:

.heading { font-size: 3rem; }

The two headings now have the same size:

rem doesn't base on the element's parent font size. It based on the root element's font size, which is the html element.

If you don't specify a font size for html, the default value is 16px in m any browsers.

As I inspected the headings in dev tools, switched to computed tab, font size of the headings is 48 (16 * 3).