Why does a Tesla car use an AC motor instead of a DC one?
You're asking about the technical tradeoffs surrounding the selection of a traction motor for an electric vehicle application. Describing the full design tradespace is far beyond what can reasonably be summarized here, but I'll outline the prominent design tradeoffs for such an application.

Because the amount of energy that can be stored chemically (i.e. in a battery) is quite limited, nearly all electric vehicles are designed with efficiency in mind. Most transit application traction motors for automotive applications range between 60kW and 300kW peak power.  Ohms law indicates that power losses in cabling, motor windings, and battery interconnects is P=I2R.  Thus reducing current in half reduces resistive losses by 4x.  As a result most automotive applications run at a nominal DC link voltage between 288 and 360Vnom (there are other reasons for this selection of voltage, too, but let's focus on losses).  Supply voltage is relevant in this discussion, as certain motors, like Brush DC, have practical upper limits on supply voltage due to commutator arcing.

Ignoring more exotic motor technologies like switched/variable reluctance, there are three primary categories of electric motors used in automotive applications:

Brush DC motor:  mechanically commutated, only a simple DC 'chopper' is required to control torque.  While Brush DC motors can have permanent magnets, the size of the magnets for traction applications makes them cost-prohibitive.  As a result, most DC traction motors are series- or shunt-wound.  In such a configuration, there are windings on both stator and rotor.

Brushless DC motor (BLDC): electronically commutated by inverter, permanent magnets on rotor, windings on stator.  

Induction motor: electronically commutated by inverter, induction rotor, windings on stator.

Following are some brash generalizations regarding tradeoffs between the three motor technologies.  There are plenty of point examples that will defy these parameters; my goal is only to share what I would consider nominal values for this type of application.

- Efficiency:
Brush DC:  Motor:~80%, DC controller: ~94%  (passive flyback), NET=75%
BLDC: ~93%, inverter: ~97% (synchronous flyback or hysteretic control), NET=90%
Induction: ~91%: inverter: 97% (synchronous flyback or hysteretic control), NET=88%

- Wear/Service:
Brush DC:  Brushes subject to wear; require periodic replacement.  Bearings.
BLDC: Bearings (lifetime)
Induction: Bearings (lifetime)

- Specific cost (cost per kW), including inverter
Brush DC:  Low - motor and controller are generally inexpensive
BLDC: High - high power permanent magnets are very expensive
Induction: Moderate - inverters add cost, but motor is cheap   

- Heat rejection
Brush DC:  Windings on rotor make heat removal from both rotor and commutator challenging with high power motors.
BLDC: Windings on stator make heat rejection straightforward.  Magnets on rotor have low-moderate eddy current-induced heating
Induction: Windings on stator make stator heat rejection straightforward.  Induced currents in rotor can require oil cooling in high power applications (in and out via shaft, not splashed).

- Torque/speed behavior
Brush DC:  Theoretically infinite zero speed torque, torque drops with increasing speed.  Brush DC automotive applications generally require 3-4 gear ratios to span the full automotive range of grade and top speed.  I drove a 24kW DC motor-powered EV for a number of years that could light the tires up from a standstill (but struggled to get to 65 MPH).
BLDC: Constant torque up to base speed, constant power up to max speed.  Automotive applications are viable with a single ratio gearbox.
Induction: Constant torque up to base speed, constant power up to max speed.  Automotive applications are viable with a single ratio gearbox. Can take hundreds of ms for torque to build after application of current

- Miscellaneous:
Brush DC:  At high voltages, commutator arcing can be problematic.  Brush DC motors are canonically used in golf cart and forklift (24V or 48V) applications, though newer models are induction due to improved efficiency.  Regnerative braking is tricky and requires a more complex speed controller.
BLDC: Magnet cost and assembly challenges (the magnets are VERY powerful) make BLDC motors viable for lower power applications (like the two Prius motor/generators).  Regnerative braking comes essentially for free.
Induction: The motor is relatively cheap to make, and power electronics for automotive applications have come down in price significantly over the past 20 years.  Regnerative braking comes essentially for free.

Again, this is only a very top-level summary of some of the primary design drivers for motor selection.  I've intentionally omitted specific power and specific torque, as those tend to vary much more with the actual implementation.