A number of companies are developing electric power plants for aircraft, basically for powered gliders and small LSA or ultralight aircraft. In most of these designs the power plant is of secondary importance and the aircraft can fly, or better, glide considerable distances without an engine purely by its basic aerodynamic properties.
Developments are in progress to electrify some every day general aviation class aircraft or even new designs. Because of many limitations to overcome, large scale applications will not come any time soon. The main problem is energy storage: liquid fuel (Mogas, AVgas or Jet fuel) has the highest energy density per weight, which is very hard to beat, except for nuclear fission or fusion. This will limit the range of any aircraft trying to fly any considerable amount of time or distance with a worthwhile payload.
For an aircraft to be able to fly, the engine/propeller must produce enough thrust to accelerate the aircraft to lift-off speed and climb for a maximum of five minutes at full power (usually). Power is reduced to maximum climb power within those five minutes until the cruise altitude is reached. After that the engine is normally set between 55 to 75% power and the aircraft flies to its intended destination.
How the propeller is driven or rotated doesn't really matter, this can either be a piston or turbine engine. Recently, a number of manufacturers are experimenting with electric propulsion systems, which include hybrids too.
The main challenge for electrical flight is where and how to store the required electrical energy for the entire flight. Or find a novel way to replenish the used energy while flying.
Dr. P. Mason debunked Elon Musk's electric VTOL jet aircraft 'idea' by explaining the limitations of batteries and why pure electric powered large aircraft will never come to be, in this video: Tesla's electric airline fail.
Compared to a piston engine the electric motor has the advantage. It is small, contains no reciprocating but only rotating parts, which is comparable to a turbine.
Among others, the Chinese firm Yuneec (pronounced as: Unique) has developed an electric motor for aircraft delivering some 40 kW (54 hp) and it weighs about 20 kg. It is used in one of their aircraft and with one 85 kg, 133,2 V 100 Ah battery (13,32 kW) it can fly some 1,5 to 3 hours (this depends on configuration, weight and such), cruise is a whopping 52 kts!
Being a glider you may expect more time from one battery charge, see image. (updated: early 2013)
Battery life is expected to be around 1500 - 3000 hrs, but real world experience shows that these items have a much shorter lifespan due to varying (fast) charge/discharge cycles, resulting in chemical and thermal stresses inside of the battery. Replacement cost is expected to be around 15000 usd.
Lets make a hypothetical short one hour flight with an aircraft powered by a Rotax 912S engine with 100 hp. During takeoff the engine runs at maximum power (100 %) for five minutes and then cruises one hour at 75 %. The amount of energy for takeoff is
(74,6 / 60) * 5 minutes = 6,2 kW or 22,4 MJ and for the cruise its: 100 hp * .75 = 75 hp => 56 kW or 201,4 MJ. In total this flight uses 223,8 MJ.
Of course, the energy efficiency of an internal combustion engine is on average about 30% and electric aircraft (85% efficient) would need a third of the 223,8 MJ which is around 88 MJ to do the same.
Conversions: 1 J = 1 W.s (1 W = 1 J/s) and one (1) kilowatt per hour is 3600 kJ of energy (3,6 MJ) and 1 hp is 746 W or 746 J/s
An 100 Ah Lithium Polymer or Lithium Iron Phosphate (LiPo, LiFePO4) battery of 133,2 V contains some 48 MJ of energy and weighs around 85 kg, not including the controller (~10 kg) (source: Yuneec). We would need two of these batteries at 160 kg to fly our hypothetical flight. In fact: Pipistrel uses two 60 kg batteries to be able to fly one (1) hour @ 70 kts (21 kWh).
If we would take 120 kg of fuel (160 L mogas) then our Rotax 912S can fly around for 9+ hours at 17 L/h with 75 % power (56 kW for 9 hours is 504 kWh of energy). Flying 9+ hours with the Pipistrel Alpha Electro or Velis will need 120 kg battery per hour of flight times 9 equals 1984 kg plus cables and hardware.
As such you may conclude is that the energy density of batteries are much lower than of liquid fuels and therefore you will see only small two-seat full electric aircraft with small battery packs with and endurance of around an hour or something like that.
Refueling our Rotax 912S aircraft with 160 liters of mogas will take around ten minutes. Recharging our 1984 kg LiPo battery with 680 MJ of energy is going to take somewhat longer. The amount of power is 680 MJ divided by 3,6 MJ (1 kWh) is 189 kW. Running a charger from the mains power (230 V) at 16 Amps (3,68 kW) will take some 51 hours. Probably a bit longer as there are efficiency losses in the charger and battery.
To reduce the recharge time you will need a three phase mains supply to recharge the batteries. On top of that, if you want to keep flying an second battery pack will be necessary. Supplemented with solar cells on the roof of the hanger (and a full day of clear skies). But I doubt if you will earn the investment back (no, not subsidized of course).
Batteries recharging in flight by solar or photo-voltaic cells is still far, far away from being a real solution for real aircraft. Small experiments with very light weight and low powered aircraft are being conducted but they are not yet suitable for the market. The limit is in physics and as solar cells ideally generate about 200 to 300 watt/m2, wing area is not nearly enough, let alone not pointing towards the sun or flying at night, clouds...