Just use the shortcut CTRL+SHIFT+P to open a new private window.

Use this link for more information Private Browsing - Use Firefox without saving history

Is there any indication in the browser which meant , the browser in private mode.

The "mask" should be there when you open a Private Browsing window, but not when you use permanent Private Browsing in Options > Privacy -> History = Firefox will: Use custom settings for History = Always use Private Browsing mode.

You can use an extension to get Private Browsing mode support per tab.

• Private Tab: https://addons.mozilla.org/firefox/addon/private-tab/
• Private Browsing mode tabs get a dashed underline
• You can toggle Private Browsing mode on/off via the right-click context menu of a tab