Can electric bike controllers be used with a brushed DC motor taken from an electric lawnmower?
Brushless motors require controllers that are completely different from controllers for brushed motors.

For any brushed motor:


Select a controller for brushed motors
The controller voltage rating, the motor's maximum operating voltage and the battery voltage should all match. Some variations are acceptable, but need to be understood.
The continuous output current rating of the controller must be not less than the continuous motor current rating.
The controller should have an overload protection adjustment that can be set for the continuous current rating of the motor.
The controller should have a short-term current limit adjustment that can be adjusted to max the short-term current rating of the motor. That will probably be something like 150% of the continuous rating for about 1 minute.
If the motor has a separate field winding, the controller must have a separate output for that. It needs to supply the proper voltage and sufficient current. If the controller has that output and you don't need it, no problem.
The controller may have other inputs and outputs that you may or may not need. For most brushed motors, you don't need speed feedback or anything like that for simple use.
The controller will need a speed setting input that is compatible with the device that will be used for that.
I would highly recommend buying only a controller for which you can download complete specifications, diagrams and instructions before purchase.



  Are there different types of controllers that must be used depending on the type of motor you have? Does anyone have a source that describes all the types of controllers?


In addition to brushed and brushless controllers, there are various types within each category, but the differences among brushed motor controllers are probably not too important in this case. For using a lawn mower controller on a bicycle, you should check the range of speed adjustment and acceleration control. We don't usually recommend specific sources of information or products. There is a lot of available information about different controller designs and features. You should study information from bicycle speed controller manufactures. Search the internet for information about unfamiliar terms that you find there.