What is the difference between a solar system battery and a normal battery?
Batteries have a lot of properties. For example, some batteries need to be fully discharged before charging. Some batteries (car starter batteries) are designed for supplying short bursts of high current and then be recharged and are not designed for a deep discharge. Usually the "usual" battery is a plain old car starter battery and the "solar" battery is a deep-cycle lead-acid battery which is more or less a variation of a car starter battery.

What you obviously will want is a battery that allows for partial discharge and allows for deep slow discharge - those are deep cycle lead-acid batteries, Ni-MH batteries and varieties of Li-Ion batteries (people say that shallow discharge is best for Li-Ion, but in professional power tools they work quite well with deep discharge unless it goes beyond some super low level and the latter is guarded against with control electronics).

The three will largely vary in price, weight per stored energy, discharge capabilities, charging requirements and control electronics requirements. Unless you need your power supply to be very portable (like for carrying in a bag) the usual choice is a deep-cycle lead-acid battery - it is relatively cheap and provides decent performance. A very important property of lead-acid batteries is they can't be stored partially discharged - that leads to sulphation and permanent damage, so you'll  have to take care of the battery being fully charged after each discharge.