Why do forces/pressures/electrons/temperatures/etc equalize? What causes various forces/etc. like to equalize? I can understand water in a tank being affected by gravity but why do high pressure systems want to move to low pressure systems (vaccuums), electrons move to cover a surface/volume and so on?
 A: This is Newton's 1st and 2nd law. If forces don't balance something will move!
$$\sum F=ma$$
If there is a higher force or pressure in one end, then if it isn't balanced (by equal and opposite forces), the stuff will start moving. 
In a water pipe you might have high pressure in one end and low pressure in the other. 
In other words this means that a force is pressing from one end, trying to make the water move to the other end - while another but smaller force is pushing from the other end. Like a strong man and a weaker man pushing on each other. 
Who will win? Will the water move towards the end of higher or lower pressure? 
I guess it is quite obvious that the stronger man will win and the weaker man will be pushed backwards. Meaning that water will be resisted by the lower pressure, so it will slow down a bit, but it will still move towards that end. 
A: So whenever there is a property gradient (change in pressure, temperature etc.) there is an energy associated with that change.
As a demonstration, water turbines use a height change (which causes the water to rush in a particular direction); so the water speed after after the turbine is less than the water speed before the turbine, and some of that energy has been converted to electricity.
Now you can see there is an energy associated with a gradient, you can also see that if you want to maintain that gradient, you need a force (like a dam).
Generally in nature, much like the case for our hydroelectric dam, that maintaining force isn't there, so there is nothing to stop that gradient getting smaller and smaller (so the potential energy, lets call it, also gets smaller and smaller); and we see that as property gradients trying to equalise.
Why do they want to equalise in the first place? Because it requires the least effort (or energy input). Nature is lazy.
