From BEAM Robotics Wiki
A somewhat derogatory term used by builders of microcontroller based robots to describe any robot that does not incorporate a microcontroller, microprocessor or any other type of programmable digital controller.robot consist of simple sensors (such as bump switches or photocells), and a transistor or two that cause one or more of the robot's motors to be turned off or on, or to turn in the opposite direction.
The example on the top-right is a Dumb-bot that can follow a line. The wheel on one side is larger than the other. Consequently this simple bot tends to turn, rather than go in a straight line. When the line is detected, the motor driving the larger wheel is turned off. As a result, the robot turns in the opposite direction. When the line is no longer detected, the robot again turns as before, until the line is again detected. This behavior allows the bot to creep along, following the line.
Another example of interesr would be Seleno, The Electric Dog. This machine was designed and built in the earliest part of the 1900s, and thought today it would be describe by the more digitally minded as a dumbot, in its day Seleno was considered a breakthrough in guided mechanisms and proof that intelligent behavior was actually the result of relatively simple processes.
See: Are BEAM robots dumb