To expand on @Autistic's answer…
There is no need for a transmission with an electric vehicle (assuming that the speed and characteristics of the motor match the application – for applications where this isn't the case electric motors are often coupled with a gearbox to (usually) reduce the output speed of the motor to something usable) because an electric motor produces maximum torque at 0 RPM.
An electric motor is basically a short circuit – it's just a long hunk of wire wound up in a coil. The only thing that prevents a motor from tripping its circuit breaker or blowing its fuse is the back EMF (electro-magnetic force) that is generated when the motor is spinning. As a result when the motor is stopped (or when something tries to slow the motor, such as when a car is asked to climb a hill) there is initially a large inrush of current. This changing current in the motor's windings produces causes a magnetic field to build and the changing magnetic field is what generates the back EMF which resists the current rushing through the motor. So, as the motor comes up to speed the current (and torque) are reduced until an equilibrium is reached.
But that's not the case with an internal combustion engine, they have to be spinning to produce any torque at all and they typically (always?) have a relatively narrow RPM range in which they develop best torque. The transmission provides a way to match vehicle speed to the RPM range where the engine produces its best torque.