I agree with most of what Mormacil said but differ on a few points.
With enough power (Volts*Amps) then it could instantly kill someone, so it would be humane. The effects would be as, if not more, grisly than a flame throws since with enough power you could vaporize their innards causing them to split open like a potato in the microwave.
For a Tesla weapon to be useful, there would need to be a strong magnetic field to confine the electrons to travel in a specific direction. Otherwise, the arc generated by the weapon would seek the nearest ground potential -- read as dirt, water, walls. But with a helical magnetic field radiating from the weapon towards the target, the electrons in the arc would be confined by a Lorentz force.
This would also provide means of defending against the weapon, powered iron would cause the magnetic field to be distorted in wild ways, making the aim unreliable.
Conductive fluids like salt water sprayed in the air would attract the arc and dissipate some of its energy be being turned into steam, rendering the arc less dangerous.
Any conductive and grounded metal would act as a lightning rod and pull the arc towards it. So a shield or a suit of armor that was grounded would protect the wearer until the metal got too hot as the arc melted a hole through it. But, thick enough and sufficiently grounded metal could last indefinitely.
And lastly, as pointed out in previous answers, battery technology used in laptops and cell phones would not have the power density required for a dangerous weapon. But, a fuel cell and one-use chemical batteries used for missiles generate tremendous power for the length of the weapon's flight time -- 20 minutes to 2 hours. These kinds of batteries are called primary cells.
It is not unreasonable to think that either of these power sources could be used. Fuel cells would require refueling. And chemical batteries only last for a specific time period after activation. Activation, in this case, means mixing the chemicals. Both aspects provide interesting limitations for gaming.