H-Bridgge
Yes, an MCU can be used.
The attached schematic is an H-bridge.
Let's get something things outlined here:
MOSFETs are transistors. They are 3 terminal devices where larger current flows through two terminals and the third terminal a small voltage signal can control the larger current. In this case, they are used as switches.
COASTING TO A STOP:
A DC motor will coast to a stop if no power is applied and no connection is made between the motor terminals. You do this by turning the all 4 MOSFETs open.
BRAKING:
But if a DC motor has it's two terminals connected together it will brake. So if you want the motor to coast to a stop, just disconnect power. If you want it to brake, disconnect power and connect the two terminals together.
In the image, this is current path is D,B, and the blue loop. You can get this by closing either Hi1/Hi2 or Lo1/Lo2. Both combinations do the same thing- short the motor terminals and make it brake.
SPINNING:
A motor's direction depends on which way current is flowing through it. You can control this by closing either Hi1/Lo2 or Hi2/Lo1. This is path C and A in the image. Each one makes the current flow through the motor in a different direction, thereby making the motor spin in different directions.
SHORTING: You never ever ever want to have either the MOSFET pairs Hi1/Lo1 or Hi2/Lo2 closed at the same time. As you can see, this would short-circuit the battery terminals. So you open and close the MOSFET "switches" in a way so that this never happens.
MOSFET TYPES:
There are two types of MOSFETs. NMOS and PMOS. The voltages that would turn an NMOS on or off do the opposite for a PMOS. They turn on and off in opposite ways:
SWITCH....ON......OFF
===================
PMOS........LO........HI
NMOS........HI.........LO
This has a few effects, and at this point the most important of which is that it is easier to turn a PMOS on/off if it is closer to +V on the battery. Conversely it is easier to turn on/off the NMOS if it is closer the battery GND.
NMOS conducts more efficiently than PMOS, therefore you could use NMOS for all 4 transistors. The problem is that the top two NMOS are far away from the battery GND. Higher voltages will be required to turn on the top two NMOS transistors. This "difficulty" is not trivial. The additional voltage boosting circuitry needed is often more complicated than the H-bridge itself with many many performance considerations, tradeoffs, and concessions to be had.
The other way is to use NMOS for the bottom two transistors and PMOS for the top two. NMOS transistors on the bottom will conduct more efficiently than PMOS and will be easier to switch on and off. PMOS at the top will be less efficient than NMOS but will be MUCH MUCH easier to switch on and off. The voltages supplied directly from an MCU pin are enough to do it.
And you would never use PMOS for the two bottom transistors ever. Not only would they conduct less efficiently but would be harder to switch on and off.