Why is there 12v, 24v and 48v? What's the difference?

Current, the flow of electrons within a conductor is inversely proportional to the voltage or potential difference, therefore as the voltage increases the current is reduced.  To calculate the current for a given load the formula P= V × I would be used, P being power, V being voltage, I being current.

As an example say we had a known load of 1000W, we know the voltage to be used is 12V, we would transpose the formula to I = P / V
 or I = 1000 / 12 which would be 83.33 amps.

When the voltage is increased to 24V is  I = 1000 / 24 which would be 41.66 amps,
When the voltage is increased to 48V is I = 1000 / 48 which would be 20.83 amps.

Because a conductors current carrying capacity is limited by its cross sectional area, or gauge using a higher voltage allows more energy to be transferred. Let's use a nominal gauge of 10 AWG which is rated at 30A, using the same formula P = V × I
At 12V the power allowable  would be 12V × 30A = 360 watts
At 24V the power allowable  would be 24V × 30A = 720 watts 
At 48V the power allowable  would be 48V × 30A = 1440 watts 

Therefore it's more economical to use a higher voltage when dealing with higher loads because the conductors and other devices, such as overcurrent protection can be smaller, thus saving in ballance of system costs.

This rationale applies to the charge controller as well, they are rated by current but allowing different nominal voltages, using a low voltage limits the array capacity in the same way as the conductor example above.

When building a system it's the load capacity which determines the most practical nominal voltage, 12V is generally considered suitable for low power systems of 1000W or less. 

Batteries follow the same rules, to achieve the current needed for a high power low voltage system, there often needs to be many batteries in parallel which is impractical as it becomes difficult to keep them in ballance leading to premature failure, not to mention the cost of conductors needed to do so.

The same formula P ÷V × I applies to batteries, 12V at 400Ah is 4800W, 24V at 200Ah is 4800W, 48V at 100Ah is 4800W, 

This is a simplified explanation which I hope helps you understand, if not ask more questions.

Couple simple points:
12V is for small, simple systems with typically less than 800 watts of panels.
48V is for full time off gridders - typically using more than 1600 watts of panels. 

Wiring runs cooler with less resistance at higher voltage levels. So 48V wiring can be ~ 1/4 the size of 12V wiring. Assuming, for example, that both systems have the same wattage flowing through them.

Also - running eight 6V batteries in parallel (for 48V) is potentially problematic. 48V allows one to run these eight batteries in serial. 

I'd like to see a 120V system but do not expect it. It would carry a potentially lethal shock hazard. Anything below 30V is considered "safe". The advantage would be found in efficient wiring and a much cheaper inverter. All the inverter would have to do is convert DC to AC.