The Dark Side of the Force is very different than the Light Side. The Dark Side uses the raw emotional power of anger, hatred and fear to give them powers far beyond your typical Jedi (who uses more harmonious feelings of peace, including apathy, to power themselves).
The Rule of Two refers to the Sith method of tutelage. The Jedi were fans of using a temple or academy system where many candidates would join. The Sith ended up rejecting this, though they did use an academy setting for sometime to train their apprentices in the ways of the Dark Side. Ultimately, the Sith felt this led to a watering down of power. Others could weasel their way through the ranks, putting weaker Siths into power and overall making the Sith be ruled by those who were weaker than they should be in the powers of the Dark Side.
This is what gave rise to the Rule of Two. The Rule of Two states that there is always two Siths, one who has mastered the powers of the Dark Side and one to crave for that power. The Master and the Apprentice structure was born. The idea is that ultimately when the apprentice Sith is ready to become a Master Sith, he will rise up and kill his master. If he can kill his master, that means he will have surpassed his master in the ways of the Dark Side, thus meaning that the Sith will always be ruled by the most powerful. The newly annointed Master will then go about finding his apprentice, and thus the cycle of the Rule of Two continues.
Darth Bane was the first Sith to really push for this Rule of Two. Darth Bane actually went so far as slaughtering every single member of the Sith Academy, ensuring that no weak user of the Dark Side would ever exist as a Sith. However, there are times when the Rule of Two has not always worked in the Sith’s favor. One Sith Master has feared that his apprentice was just waiting around until the master died of old age before taking the title, and the master lamented this meant that the Sith would be ruled by a weakling who could not take power for themselves. Luckily, (in the Sith kind of way) he was later killed by his apprentice and felt the rush of pride in his apprentice.
However, Darth Sidious who would later become Emperor Palpatine, sought to disband the way of the Rule of Two, same with his master Darth Plaguesis. Darth Plaeguesis dreamt of an entire Sith academy again and sought eternal life so as to see his dream to come true. But his apprentice Darth Sidious had other plots in mind. Palpatine also wanted to end the Rule of Two, in favor of the Rule of One, where he would live forever as the only true power of the Sith.
This is why Emperor Palpatine killed Darth Plaguesis, and then made sure Anakin Skywalker became mortally wounded before turning Anakin into the half robotic Darth Vader. In his weakened state, Darth Vader could never hope to achieve the same powers of the Dark Side as Emperor Palpatine did. This was the securing of the Rule of One, where Palpatine became the ultimate and uncontested ruler of the Dark Side and of the Sith.
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