Your default search engine refers to the search engine used when performing a search using the Address bar (also referred to as "Address bar", "Location Bar", "Omnibox") located at the top of your browser:

You can tell at a glance which search engine you currently have set at default by the symbol displayed on the left side of the Address bar. In the above example, I have Qwant set as my default search. When a new default search is selected, you can see the change will be reflected in the icon.

As you can see below, I've changed my default search engine from Qwant to DuckDuckGo and the left hand side of the Address bar now displays the icon for DuckDuckGo:

There are several ways to switch what engine is used when performing a search in the Address bar.

Set Default Search in Settings

Using the following method, you can adjust the  Address bar's default search behavior. This setting will remain configured until it is modified so that any time you perform a search it will be run through the engine selected:

1. Open the Menu 
2. Navigate to: Settings —> Search Engine

Here, you can select the default search engine for both normal and private browsing windows:

Here you may adjust other search options as well:

• Improve search suggestions — Toggle this feature on/off to send search information to the default search engine you're using in order to get better results/suggestions. Note that this option is off by default as well as automatically disabled when using Private browsing.
• Web Discovery Project — Toggling this option on will anonymously contribute some of your search/browsing data to refine Brave Search. Learn more about the Web Discovery Project here.
• Index other search engines — Toggling this option on allows the browser to index any search engines used so that they can be added to/managed in the browser at a later time. This will apply to any search engine that follows the OpenSearch specification. 

Manage Search Engines

Selecting this option will display a larger menu with all of your available search engines. The Search Engines section lists all engines you can set as your default right now. This list may be pre-populated with several common domains:

To change your default search engine from this page, click the "three dots" icon on the right-side of the entry and select Make default — note that you can also remove the entry from this menu as well by clicking Delete.

The Site Search section lists search engines that you may have used while browsing (assuming the `Index other search engines` option is enabled) and allows you to search a specific site. Additionally, you can also add and configure search shortcuts for internal browser pages, such as your Bookmarks or History page:

Clicking the "pencil" icon allows you to edit the name of the search engine, the URL as well as the quick access shortcut that is associated with the entry:

The Inactive Shortcuts section displays search engines that have been indexed automatically via open search as discussed previously. Note that in order to be able to use these, you will need to click the Activate button on the entry — once done, it will be added to the Search Engines section at the top of the page:

Adding Search Engines
If your preferred search is not on your list or indexed automatically, you can add it manually. Navigate to Manage Search Engines as you did above and click the Add button. You'll see the following text boxes:

• Search Engine
  What you would like to name the search
• Keyword
  The keyword associated with the search [see Using Keyword Searches]
• URL with %s in place of query
  This is asking for the desired search engine's  web address but replacing the search query texts with "%s". For example, if I wanted to Add the Reddit search to my list, I can navigate to Reddit and perform a search. Note the highlighted text in the URL:

The highlighted text is the search query itself. That is, for Reddit, every search term entered will appear after the q= text. This is the text that you want to replace with %s. The resulting URL would look like this:

After clicking save, your new search will appear in the Other Search Engines list for you to use.

Note: In addition, search engines can also be added to your list simply by navigating to the desired search address and performing a search. You will see the new search reflected in the Other Search Engines list.

Using Keyword Searches

You can also switch the search engine being used with Keyword Searches. A Keyword is simply a shortcut typed in to the Address bar that tells Brave that you'd like to perform a search using a specific site associated with that keyword. Once the search is performed, Brave will revert all searches performed in the Address bar back to the default search engine configured in settings.

For example, if my default search engine is set to DuckDuckGo, but I'd like to search for a particular item on Amazon, I could input:

:a [space or tab] [my search query] 

Doing this changes the search to Amazon.com as shown above. You can enter your own preferred keyword when manually adding a search engine to your list [Adding Search Engines]. Common practice is to prepend the search keyword with a colon " : " followed by a letter(s).

Alternatively, you can also enter the the domain that holds the search you'd like to use and hit [Tab] to achieve the same effect:

Note that you can change the command used to initiate the quick search at the top of the Manage Search engines page: