At one end of the scale, there's a pure ICE--Internal Combustion Engine. This is the normal gasoline or diesel engine everyone knows about. At the other is pure electric--Tesla, Rivian, etc. A hybrid is a mix of the two.
Why add the complexity of a hybrid?
Conventional brakes just turn speed into heat. Hybrids have regenerative braking--this runs the generator, slowing you down as it charges the battery so you can use that charge to get moving again. ICE engines are generally most efficient at a particular speed...but that speed is rarely the one you want to use for normal driving. They are often particularly inefficient at speeds you need to use most often. Atkinson Engines are an extreme version of this--They tend to make less power per liter/cubic inch of engine size, do more work per gallon of gas, and by themselves do not have enough power for normal driving at low engine speeds. Conveniently, electric motors are best where the Adkinson is worst, so a blend of the two works well. Hybrids also make it more practical to power accessories with electricity so the engine doesn't have to run to keep the AC on for example.
A mild hybrid system has a relatively small electric system. It recovers power from braking, but the hybrid motors and batteries may not be capable of powering the car alone. An early BMW system merely charged the normal 12v battery to reduce the load of the alternator. The first generation Honda Insights had "Integrated Motor Assist" a 13 hp electric motor sandwiched between the 67 hp ICE and the transmission. This allowed regenerative braking and gave a bit of extra power to the ICE--but the engine had to be running if the IMA was running. Honda continued using IMA with bigger electric motors until fairly recently. There isn't a clear line between mild and full hybrid.
A full hybrid will have a bigger electrical power system, generally with the capability of running on pure electric for short periods. The Ford Maverick pickup I have on order has a 162 HP Atkinson cycle ICE, plus 131 hp electric from two motor-generators. Electric motors have peak power at low speeds, while the ICE peak is at a higher speed. Since the peaks don't align the total is lower than just adding the two, in this case 191 hp. The ICE will attempt to only run at it's most energy efficient speed. If that's not quite enough power, the electric can help. If that is too much power, the extra goes back into the battery. When the battery nears full, the ICE will shut off, and the electric motors will take over. Under hard acceleration both ICE and electric will run until the battery is depleted, then it will be ICE alone with reduced power. Note, the electric motor-generators on the Maverick are always used even when battery power isn't, it is the various combinations of "motor" and "generator" that "shift" the transmission. In some circumstances one motor generator will generate electricity and send it to the other motor generator to keep the ICE at it's optimum speed and load. Reverse is entirely electric.
A plug in hybrid has more batteries than a full hybrid with the ability to run some substantial distance on battery alone. It is intended to be plugged in and run off that charge for most trips, but with the ability to use gasoline for longer trips. This may be similar to a full hybrid with bigger batteries, or closer to a full electric, with a relatively small ICE that only charges the battery with no connection to the wheels.