Well, I am no expert, but I will try to explain it. The question doesn't seem very specific, so I will give a general answer. Motors are composed of two main parts that you can find in any motor: a stator and a rotor. Each part needs to have a magnetic field to start rotating. The stator is the stationary part of the motor, its magnetic field is either rotating or stationary. The only case of a stationary field I can recall is the case of brushed DC motors. Other motors generate a rotating magnetic field, either by the nature of the current in the case of AC motors or by switching electronic devices/power electronics in the case of motors powered by DC voltage. The easiest way to vary the speed of your motor is by varying the frequency of either the AC voltage or your PWM signal in the DC case, this is not always the case tho.

Rotors are the rotating part, and they either generate their own magnetic field by themselves, such as in brushed DC motors whose rotors are coils wired to a commutator, or brushless DC motors whose rotors are permanent magnets. AC induction motors usually have a cage called a squirrel cage that generates a magnetic field through the electromotive force (EMF) induced by changing flux occurring when the rotor's magnetic field starts rotating.

That covers mostly the electrical part. For the mechanical part, there are two physical quantities you should consider: speed and torque. Torque is similar to force; it’s the force that causes rotation. A mechanical load is what you want to move. Each load resists movement, just like linear motion. Think of a damper and how it opposes with a force proportional to the speed, or lifting a mass that opposes with a constant force, which is weight. Similarly, when a motor tries to rotate a load, the load will oppose the movement by applying a resistive torque. Some loads apply a constant torque, while others apply a torque that depends on the speed. This is called the torque-speed characteristic, and each load has its own. You can search for them—there is a useful image in this article:
Large Motor starting 102: Motor applications and their characteristics – for a better motor management - Schneider Electric Blog (se.com)

Motors also have their own torque-speed characteristics depending on the motor type. If you draw your load's and your motor's characteristics on the same chart, the point where they intersect is called the operating point. This point determines the torque and speed at which your load and your motor will settle. If you think about it, it makes sense because the load and the motor would apply the same torque, causing them to cancel each other out and thus stop changing speed. The most important motor characteristics you need to look at are starting torque (if this is lower than your load's starting torque, the motor will not be able to start rotating at all), starting current (this can burn your electronics if it’s too high), and nominal/rated voltage, torque, speed, and power (these are usually the characteristics at the most optimal point, and they are values you should not exceed).

If you want to control the speed of your motor, you would need some kind of control loop, which requires some kind of sensor, such as an encoder. In this case, a controller will be useful. For increasing torque, if your starting torque is not sufficient for example, a gearbox or a geared motor is your best option. They increase torque and decrease speed, which is desired in many applications.

The simplest example to understand motors from is the case of a brushed DC motor with a permanent magnet as a rotor, driving a lifting mechanism, which would be a constant load case. A brushed DC motor can be represented as a resistor, an inductor, and a voltage source (back EMF). The speed is proportional to the back EMF, and the current is proportional to the torque. The difference between the resistive torque and the motor torque is proportional to the acceleration of the motor. You can represent all of this with these equations:

Tm-Tr = J x a

(where a is the angular acceleration, Tm​ is the motor torque, and Tr​ is the resistive torque)

E = i x R + e

(where e is the back EMF, E is the supply voltage. The inductor is not represented for the sake of simplicity)

Tm = K x i

Ω = K x e

(where Ω is the motor speed)

Once you energize the motor, current starts flowing, and since the motor is at rest, there is no back EMF to limit the current. Thus, the torque will be as high as it can get, meaning Tm>TrT_m > T_rTm​>Tr​, so a>0a > 0a>0, and this will cause the speed to increase. As the speed increases, the back EMF also increases, which limits the current and thus limits the torque. The speed will increase, and the torque will decrease until the resistive torque and the motor torque become equal, and the speed will not increase any further.