A device which is normally a stripped down, low power MicroProcessor. In robotics - you will often come across these.
They normally have many inputs and outputs which can be interfaced directly with devices (possibly they may need relays/power transistors). They often include DigitalToAnalogue and AnalogueToDigital converters as well as PWM motor controllers and Servo Motor channels. You can purchase single chips, boards to whole them, or complete development kits.
All of these controllers have IO pins (or GPIO) for controlling external devices, and may have ADC’s to input from analogue sensors. They are mostly capable of PWM - a method of using timing on a digital signal to control an analog output.
The Raspberry Pi and Beaglebone are both Linux computers miniaturised. So any language you could run on linux will do. You will probably need to write the code onto an SD card for the pi and may need a screen and keyboard.
The Arduino has a well known IDE and tool chain around C++ packaged in a surprisingly easy environment, and needs only a USB port. It has lots of library support and the internet will have example code or libraries for almost any sensor you can think up. It can be used to program the ESp8266 and ESP32 instead of the other interpreters.
The Bare AVR requires a bit more with specific programming cables. An Arduino can be modified into an AVR programmer.
Bare ARM devices come from a multitude of vendors with a variety of capabilities. Most are well supported in terms of compilers and tool chains, mostly in C/C++. Some have higher level interpreters using micropython or Lua. Specialist programming devices may be needed for this.
The 8051 is a somewhat venerable processor type, but are often present on quite modern controllers - and have great support in terms of compilers.
This page listed controllers previously that are less used currently, but preserved here for context.
For a comparison table on these devices - go to microcontrollers roundup.
It is not unusual to find that in addition to buying a microcontroller development board, a programming cable and other supporting hardware, that you will also need a development environment.
The minimum is a compiler to actually build the code, and then a programmer to download the code onto the board/chip. Often, there are a number of vendor ones, and also commercial ones. There are also free ones. The vendor and commercial development packages range from £50 to thousands of pounds.
Free/Open source tools:
More tools are covered on the family specific pages above.
Time to build your own robot! Using a Raspberry Pi with parts and some time, you can use my book to learn how to make and program a robot with automatic behaviours.