Some German messages were encoded twice, using different settings each time. These were the German Navy's 'Offizier' and 'Stab' ('officer' and 'staff') messages, used for important strategic messages. The different naming schemes reflect who the messages were sent to; 'staff' messages were sent between headquarters, while 'officer' messages were sent from headquarters to commanders at sea and vice versa. The two systems worked similarly. The procedure for encrypting an 'officer' message started with a list of 26 settings. These were printed with a letter of the alphabet beside them, and were changed every month. You selected the setting you wanted to use, and noted down the letter that went with it. You set up the Enigma machine using the given settings, and encrypted your message. To ensure that the recipient knew which setting you had chosen, you picked a name starting with the letter that corresponded to your settings. This was appended to the enciphered message. The whole combination was then enciphered with the standard Enigma settings for that day. This could then be transmitted. The 'staff' setting used a different set of initial settings, but otherwise worked the same.
The double encipherment was a much more time-consuming and complex than just using a single encipherment. This meant that there was a greater risk of mistakes that would render a message junk. The method was also not as secure as it seemed. As the settings for the initial encipherment were only changed every month, they were often repeated. If the Allied codebreakers were able to break an 'officer' or 'staff' message, then they could use those settings to read any other message sent using them, as long as they'd broken that day's general Enigma settings.
As far as whether or not the Germans knew about the Enigma's behaviour with respect to enciphering a letter as itself, they did but didn't consider it a weakness. For a message written in standard German, with a wide distribution of letters, a message enciphered in Enigma would still look like a random string of letters. The weakness would only be apparent if either the message had a large amount of a single letter, or if the enemy codebreakers had some idea of what the message might contain (a 'crib', in the language of Bletchley Park). Neither of these would be the case under an ideal system. The problem was that the German (and Italian) system was not ideal. Signals procedures were often fixed, and radio operators often lazy, meaning that messages were formulaic; this offered many cribs for Allied codebreakers to exploit. Some radio operators, when told to send a meaningless 'dummy' message to confuse Allied listeners, sent messages that were long strings of a single letter. Such messages could easily be broken, and the Enigma settings obtained.