For the Address Bar

1. Type about:config in you search bar as below

   

2. You'll then have a humourous notification box come up in the middle of the browser telling you the following:

   What you'll be agreeing to is that you'll be able to modify values of certain Firefox variables which change the overall user experience (so I strongly recommend you don't modify anything you don't need/know about).

3. Once you've clicked on "I'll be careful, I promise" the following search bar and list will come up:

   

4. Within the searchbar look up keyword and you should have the entries in the screenshot below

   

5. Lastly right click on keyword.url and click Modify and change it to whatever search engine you want, the example I've given here is DuckDuckGo (replace DuckDuckGo with any other search engine of your choice):

   

For those interested/wondering the ?q= at the end of the URL you post (as far as I know) is just how search engines determine the query required to be sent to the search engine.

For the search bar on the right

Just click on the triangle to the left and a drop down list will give you a choice of what search engine to use.

If your search engine isn't already there you will need to add it manually by clicking on the Manage Search Engines option at the bottom of that list of search engines and it will open up the following window:

Once there you simply click the Get more search engines... URL in orange and you can choose from a wide list which sites to add to your list so you can have them in your right toolbar at the ready.

Hope this answers your question.

Edit: I recommend users have a look at Dan's answer below, not exactly what was asked but still a very useful feature, well worth a read.

&t=

Through partnerships with developers and companies, DuckDuckGo has been integrated into many applications. In these partnerships, a portion of DuckDuckGo's advertising revenue is sometimes shared back. To assign advertising revenue and collect anonymous aggregate usage information, developers add a unique "&t=" parameter to searches made through their applications.

If you want to remove it, edit the file:

/usr/lib/firefox/distribution/searchplugins/locale/*language-here*

Edit all the languages just to be sure.

Edit for 20.04

1. Open Firefox, navigate to Preferences --> Search, and remove the per-installed DuckDuckGo search engine
2. Download the search engine new from the Firefox Add-ons directory. Select this as the default search engine in the preferences The string, &t=canonícal, ceased to insert itself into the searches via the address bar thereafter. With help from Komalleana Byzantis

Below are before and after pictures:

^{Source:DuckDuckGo Forums}