As the name suggests, this technique is simply a very quiet method of killing an opponent, usually from behind with a blade. Additionally, because the user moves silently, the attack cannot be anticipated and defended against. The user will often slit their opponent's throat, preventing any cry that might give them away. The Seven Ninja Swordsmen of the Mist are shown to be masters of this technique, and were even good enough to track and kill opponents through sound alone. Some shinobi also opt to use a cover such as mist or even frost to rob their opponents of their field of vision and then attack them.
Although the name of this technique is written in kanji (無音殺人術), it's pronounced in English ("silent killing" (サイレント キリング)). The Japanese pronunciation of the kanji would be Muon Satsujin Jutsu, which would literally translate to "soundless murder technique".
According to Kakashi, this technique allows the user to kill their opponent so quickly, that they die before realising it.
Silent Killing may have been inspired by various Chinese and Japanese books released during the centuries detailing ways on how to stealthy, and silently kill another person using various weapons and by targeting vital organs in the body.