The procedure of setting Conkeror as your default browser varies by OS, and sometimes by program.
General information on the subject can be found here.
For more detail about launching conkeror including opening new tabs in existing instances see Invocation
First off to allow selection of Conkeror as browser there needs to be a .desktop file. On debian this file contains insufficent mime types for the browser to be a possible target for urls. To fix this we need to create a new .desktop file. It could be placed in /usr/share/applications/ but if the file already exists then it will be overridden by OS upgrades so instead we are going to add it to your home directory.
first make sure the directory exists
mkdir -p ~/.local/share/applications/
then edit the file ~/.local/share/applications/conkeror.desktop and put the following contents in
[Desktop Entry] Version=1.0 Type=Application Terminal=false Icon=conkeror.png Name=Conkeror Comment=Emacs like Web Browser GenericName=Web Browser Exec=conkeror %u Categories=GNOME;GTK;Network;WebBrowser; MimeType=text/html;text/xml;application/xhtml+xml;application/xml;application/vnd.mozilla.xul+xml;application/rss+xml;application/rdf+xml;image/gif;image/jpeg;image/png;x-scheme-handler/http;x-scheme-handler/https;x-scheme-handler/ftp;x-scheme-handler/chrome;video/webm;
Now that is done Conkeror can be selected as the default browser for desktop environments like GNOME and KDE.
If you are not using one of the major Desktop environments like Gnome or KDE and you still use xdg-open you can override the default xdg browser with xdg-mime.
xdg-mime default conkeror.desktop x-scheme-handler/http xdg-mime default conkeror.desktop x-scheme-handler/https
To set it as the x-www-browser which may be used by some applications in Debian and Ubuntu to find the default browser.
sudo update-alternatives --config x-www-browser
For details about manual configuration of Emacs see the EmacsWiki