Tesla does use induction motors where virtually all other electric car makers use permanent magnet synchronous motors. The two types work very similarly and there are advantages and disadvantages to both.
It is mainly historical that Tesla uses induction motors (as well the fact that Tesla invented the induction motor over 100 years ago). Teslas started life using motors and controllers from AC Propulsion that happen to use induction motors.
All motors are AC at some point (except for homopolar motors). The differences are where the conversion from DC to AC occurs.
In a conventional DC motor the conversion is mechanical in the commutator. It has the disadvantages of limited lifetime, sparking, efficiency and also is not as amenable to computer control.
When Induction motors or Permanent Magnet Synchronous Motors the conversion from DC to AC happens in an electronic inverter that gives a great deal of control over the speed, torque and regeneration in the car.