Economist comes with two different sidebar features - one with simple text - and the other with an image and a title link.
Let's see how both are created.
1) Firstly, both are widgets - so we go to Appearance > Widgets - 'Pages Sidebar' (if a standard page) or just 'Sidebar' if it is on a Blog post or blog archive.
Let's modify the simpler, text one first which looks like this:
2) When we open up the widget it looks like this:
Title field: Left blank
Content field:
<h3>More Information</h3>
<p>Tim Peters<br/>
CEO & Managing Director,<br/>
Economist Ltd.</p>
<p>+27 834 25 424<br/>
<a href="mailto:info@yoursite.com">
tim@economist.com</a></p>
CSS Classes field: page-aside-box-shadow
So, it's just a matter of changing this text to your own, simple enough.
3) Now, let's do the feature with an image which looks like this:
When we open up this widget it looks like this:
Title field: Left blank
Content field:
<div class="inner">
<h3><a href="#">Independent advice based on proven research methods</a></h3>
</div>
CSS Classes: Left blank
So, to replace the image with your own for example, go to the 'Media' section in WordPress and upload an image you'd like to use and note the URL (path). Replace the image path above - this part:
With your own.
Also replace the # reference with your own link to a page or post.