Choosing a motor for your robot is likely routine once you determine your drive needs and the capabilities of individual motor types. Here we will explore the pros and cons of different types of motors. There are reference materials across the Internet to help you if you haven't found what you need here.
AC Single Phase
A lot has been said about DC motor types but finding notes about AC motors are not as common in hobbyist robotics. AC motors are a great tool for non-mobile or only semi-mobile robotics. The higher voltage leads to lower currents and smaller wiring. Even single phase motors can be driven at different speeds with pulse width modulation if the motor is inverter duty. Don't discount the AC motor unless you have to; the motors are often less expensive than their DC counterparts.
AC 3-Phase (ACIM)
Continuing with AC motors, we present the three phase motor. While typically used in industrial and HVAC equipment the 3-phase motor requires a three phase power hookup, which isn't typical in a workshop. The great thing about AC three phase is the ease of picking up a variable frequency drive for control and some drives can be converted to inverters (DC to AC) if you don't have the required incoming power or desire to build your own. Three phase motors can vary speed in either direction and also provide regenerative braking if desired.
Stepper motors are driven in steps by separate coils called phases. Because of the known step angle the stepper is great choice for precise movement control.
Pros: Positioning, accurate, low speed high-torque, excellent speed control.
Cons: Low efficiency, lower top end speed with low torque at high speeds, specialized motor control required.
Stepper Code Example 1 Example of Arduino Code using AccelStepper Library
Brushed DC motors are very common and are typically the kind that come in low cost robotic kits.
Pros: Cheap to free (scrap/salvage), Simple two wire motor control.
Cons: Eventually brush maintenance required (or toss and replace).
Brushless DC (BLDC)
Brushless DC motors are a less common motor that require a specific motor controller. This motor is typical in equipment such as hard drives.
Pros: Longer Life, Efficient, More Torque per watt, quiet and reliable.
Cons: Higher Cost, Special motor controller.