Is 'Browser Default' the name of something?

Type about:addons<enter> in the address bar to open your Add-ons Manager. Hot key; <Control> (Mac=<Command>) <Shift> A)

In the Add-ons Manager, on the left, select Extensions. Make sure you know what each one is and what it does.

Type about:preferences#search<enter> in the address bar. You can select any search engine to be your default from here.

You can also remove any engine from here.

Browser Default - is what the option lists. It is always Yahoo! I will delete Browser Default as an option and change it to another search engine and it will work until I completely close out of the program. When I re-open Firefox It will have Browser Default selected as the search engine and using Yahoo when I had chose another search engine prior to closing the program.

You may have ad/mal-ware. Further information can be found in this article; https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-caused-malware?cache=no

Run most or all of the listed free to use malware scanners. Each works differently. If one program misses something, another may pick it up.