that displays most visited pages and recent pages for fastening your browsing. when you typing a address it lists your recent and most visited pages.

in edit->preference->privacy you can disable saving history

and you can also controll the contents of that dropdown button by customizing location bar.

thank you

Private Browsing - Browse the web without saving information about the sites you visit

As you browse the web, Firefox remembers lots of information for you: sites you've visited, files you've downloaded, and more. There may be times, however, when you don't want other users on your computer to see this information, such as when shopping for a birthday present.

Private Browsing allows you to browse the Internet without saving any information about which sites and pages you’ve visited. This article explains what information is not saved when in Private Browsing and gives you step-by-step instructions for using it. Warning: Private Browsing doesn't make you anonymous on the Internet. Your Internet service provider, employer, or the sites themselves can still track what pages you visit. Private Browsing also doesn't protect you from keyloggers or spyware that may be installed on your computer.

Do you mean the arrow at the right end of the location bar that opens a drop-down list?

This list shows URLs that you have typed or pasted in the location bar sorted by 'frecency' (frequency and recency). Entries in this list seem harder to remove.

• https://developer.mozilla.org/en-US/docs/The_Places_frecency_algorithm

Use these steps to remove data from a drop-down list:

1. highlight an entry in the drop-down list
2. press the Delete key (on Mac: Shift+Delete) to remove it (do not click with the mouse).

Private Browsing - Browse the web without saving information about the sites you visit

As you browse the web, Firefox remembers lots of information for you: sites you've visited, files you've downloaded, and more. There may be times, however, when you don't want other users on your computer to see this information, such as when shopping for a birthday present.

Private Browsing allows you to browse the Internet without saving any information about which sites and pages you’ve visited. This article explains what information is not saved when in Private Browsing and gives you step-by-step instructions for using it. Warning: Private Browsing doesn't make you anonymous on the Internet. Your Internet service provider, employer, or the sites themselves can still track what pages you visit. Private Browsing also doesn't protect you from keyloggers or spyware that may be installed on your computer.