Uranus is what you find when you go beyond the restrictions of Saturn. Freedom and rebellion. It is a new, sudden, inexplicable page in your own personal history. It is beyond the practical logic and the material concerns. And most importantly, beyond the human-made order (governments, corporations, Saturn).
Uranus was the first of the newly discovered planets, those planets that ancient astrologers were not able to see with their naked eye. Interestingly, Uranus actually IS visible in the sky with a naked eye. Not always but still. We are able to observe it through certain parts of its orbit. Yet none of the ancient astrological traditions (Hellenistic, Vedic, Chinese) noticed it. That's because we "see" planets only when we are ready to discover their energy. We see them with our eyes only when we the energy they represent has come into our conscious realization.
Imagine what a shock it was. Philosophical, astrological and scientific traditions had for centuries believed that Saturn marked the end of the Solar System. It was the 7th planet and 7 was a sacred number in mythology. People were used to it and liked it. It creates a sense of completion and cosmic orderliness.
Now all of a sudden, outside the structured order of Saturn, a giant cyan blue ball was rolling (not spinning) in the sky. People found themselves with an 8th planetary body on their hands which they didn't know what to do with. How do we put it in the framework within which we explain to ourselves the world we live in? Hence Uranus's associations.
Firstly, it challenges you to think outside of the box, the box being the framework of the 7th initial planets. It represents what lies outside the traditionally accepted order of things or outside the structure of society. Everything that is novel, so-far-unseen, thus-far-unheard-of, that runs counter to the established traditions and is revolutionary, is said to be of the nature of Uranus.
Uranus was discovered in 1781, around the time of the people's revolutions (the American, the French) which is why it is associated with the "power to the people" philosophy and grassroots actions. 1775 and 1789 saw ordinary people coming together to take public matters in their own hands. Uranus rules over Aquarius, the sign opposite Leo (the king of the jungle, the ruler of the kingdom). The counterpart to the ruler are the ruled, so the people. So the time around Uranus's discovery saw an unprecedented rise in desire for freedom and equality which resulted in rebellions and revolutions.
The planet itself is also unlike the other planets. The most outstanding element of its construction is its unique axis of rotation - its north and south poles are where the other planets' equator is. The planet is flipped sideways and so it rotates almost horizontally on its axis, not vertically. None of the traditional planets does that so imagine how lunatic-like Uranus appeared to be. This is the basis for Uranus's association with doing things differently, with innovation, chaos, erratic behavior and eccentricity. One day on Uranus lasts a little over 17 hours meaning it revolves on its axis relatively quickly giving its energy speed and its different-ness a chaotic character. In astrological divination it represents quick, unusual, unbelievable, inexplicable events.
Uranus is the coldest planet in our system and so Uranian (or Aquarian) people often come across as distant and unemotional. It is cyan in color and cyan is associated with electricity. From there stems the association with electronics, science and technology.
Around the time of Uranus's discovery a new chemical compound was discovered and was named Uranium, after the new planet. Uranium is mostly used to power nuclear reactors and atomic bombs because it can release enormous amounts of energy (just like the enormous amount of power that lay people coming all together can release).
Furthermore, it was through the testing of Uranium that the process of radioactivity itself was discovered. Radioactivity is a process of unstable atoms becoming stable again by releasing radiation. So Uranium atoms are in a state of constant stress and instability, sizzling, diminishing themselves in attempts to regain stability. This is why Uranus is often associated with anarchy, chaos, instability and quick "flips of the script".
And lastly, Uranus rules over human rights and humanitarian endeavors because it favors and protects the power of "small" people. It rules the sign opposite Leo (the king, the ruler) and represents the counterpart to those who rule - the ruled, the masses.