There are two main traditional ways of storing electrical energy, you can use a rechargeable battery to store the energy chemically, which will store quite a lot of energy up to 350 watt hours per kilogram, but it will do so quite slowly and for a limited number of recharge cycles. The other option is to use a capacitor, this in a simple case is basically two sheets of metal with a layer of insulator in between. If you apply a voltage some charge will flow onto the sheets and will be attracted to the opposite charge on the other sheet. because they are not using chemical reactions capacitors can charge and discharge thousands of times faster than batteries and they waste much less energy as heat.
The problem is that conventional capacitors store less than a thousandth of the energy of batteries. To increase this you need to reduce the distance between the plates stabilising the charges and allowing the capacitor to store more energy at the same voltage. Recently by using activated carbon with a huge surface area filled with an ionic liquid as an electrolyte supercapacitors have got this up to and energy density about as good as lead acid batteries.
Now Chenguang Liu and colleagues has used graphene, which is essentially a single layer of graphite a single atom thick as the electrode. This is about the thinnest an electrode can get, and they have altered the graphite so that it is curved, and won't lie flat and stack. This means that the ionic liquid electrolyte can get between the layers and produces a very efficient capacitor. They can store between 80 and 136 Whrs/kg which is about as good as a nickle metal hydride battery, which were the standard batteries 10-15 years ago. That might not sound very impressive until you realise that it would charge in seconds or at the most a couple of minutes as opposed to hours for a lithium ion battery.
This would both allow an electric car to charge up as fast as you would normally fill up with petrol, and make it ideal for storing large amounts of energy very quickly such as when you brake hard in a hybrid vehicle.