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Accumulator is an energy reserve that your robot uses to power modules and equipments that require a quick burst of power.
All robots have an internal reactor, and that reactor provides the RP units (overall known as a "powergrid") for fitting purposes. Modules that are fitted to the robot will be connected to the powergrid and reserve a portion of power for it's use (so that's why you have less RP to use after fitting a module). Such module will operate using just the power the reactor generates. But that wouldn't be enough for high-power modules like guns, miners, shield generators and many others. Hence, those modules are classified as "Active" and require a portion of accumulator to do their job.
When needed, the Accumulator discharges required energy. As it is with many conventional accumulators on Earth, it's recharge speed follows a Gauss curve (also known as normal distribution), with fastest charge speeds achieved around 50% capacity while slowing down at the low and high charge.
Remote Influence
Accumulator can be affected by the entities outside the robot as well. Players and NPCs can help recharge or drain the accumulator using Energy Transferers or Energy Neutralizers\Drainers
There is also a Recharge NEXUS that can boost the recharge rate of the whole squad
Accumulator upgrades
There are several modules that can help make the accumulator more powerful. There are rechargers, additional accumulator banks and reactor sealings, made to shield accumulator from negative effects
Full description of those modules can be found in the Accumulator upgrades article
Other ways to gain charge
An Energy Injector is a great way to boost the accumulator to either keep it in the sweet spot of recharge, or gain as much energy as possible to keep going under heavy pressure. Frequently used in PVP
ERP are another way to deal with pressure, converting incoming damage to energy ready for repairing that damage. They are fine-tuned to absorb only one type of damage however, and a bad incoming damage\ERP combo can ruin the energy feedback cycle.