A microcontroller is a system on a chip. Whereas your computer is made up of several discrete components: a processor, RAM sticks, a hard drive, an ethernet port, etc.; a microcontroller has all those components built into a single "chip" or package. This makes it possible to build systems with minimal part count.

Lots of things! Microcontrollers are the central part of systems known as embedded systems. These systems are everywhere but you don't usually notice them. These systems control the brakes of your car, wash your clothes, print your documents, keep you warm, keep you cool, optimize the fuel consumption of your car, etc.

The main trait of these systems is that they operate without user intervention even if they expose a user interface like a washing machine does; most of their operation is done on their own.

The other common trait of these systems is that they control a process. And for that these systems usually have one or more sensors and one or more actuators. For example, an HVAC system has several sensors, thermometers and humidity sensors spread across some area, and several actuators as well, heating elements and fans connected to ducts.

All these application I've mentioned, you can probably implement with a Raspberry Pi, a computer that runs Linux. Why should I bother with a microcontroller that operates without an OS? Sounds like it would be harder to develop a program.

The main reason is cost. A microcontroller is much cheaper than a general purpose computer. Not only the microcontroller is cheaper; it also requires many fewer external electrical components to operate. This makes Printed Circuit Boards (PCB) smaller and cheaper to design and manufacture.

The other big reason is power consumption. A microcontroller consumes orders of magnitude less power than a full blown processor. If your application will run on batteries that makes a huge difference.

And last but not least: (hard) real time constraints. Some processes require their controllers to respond to some events within some time interval (e.g. a quadcopter/drone hit by a wind gust). If this deadline is not met, the process could end in catastrophic failure (e.g. the drone crashes to the ground). A general purpose computer running a general purpose OS has many services running in the background. This makes it hard to guarantee execution of a program within tight time constraints.

Where heavy computations are involved. To keep their power consumption low, microcontrollers have very limited computational resources available to them. For example, some microcontrollers don't even have hardware support for floating point operations. On those devices, performing a simple addition of single precision numbers can take hundreds of CPU cycles.

Hopefully, I don't need to convince you here as you are probably familiar with the language differences between Rust and C. One point I do want to bring up is package management. C lacks an official, widely accepted package management solution whereas Rust has Cargo. This makes development much easier. And, IMO, easy package management encourages code reuse because libraries can be easily integrated into an application which is also a good thing as libraries get more "battle testing".

Or why should I prefer C over Rust?

The C ecosystem is way more mature. Off the shelf solution for several problems already exist. If you need to control a time sensitive process, you can grab one of the existing commercial Real Time Operating Systems (RTOS) out there and solve your problem. There are no commercial, production-grade RTOSes in Rust yet so you would have to either create one yourself or try one of the ones that are in development.