How to produce rotating movement electromagnetically with a coil and with a permanent magnet? I would like to know this:
To produce rotational motion by eletromagnetism, there are several ways and systems of doing it, but lets say I want to create rotational motion using a permanent magnet and a coil. The fixed magnet creates a constant magnetic field, and the coil, when current goes through it, creates a magnetic field, but this magnetic field has to change directions to create rotational motion, and it happens because current must change directions as well. 
I know how it works, I've seen it, there are 2 brushes that current goes to and creates magnetic field. And there are 2 coils on the outside. So it doesn't matter if it is AC or DC, ok.
My question is because theoretically rotational motion is created if magnetic field changes direction, and I was wondering what would make it change direction.
If current must change directions, would it be used AC or DC to create rotational motion? Intuitively, It makes me think AC is always alternating directions and DC don't, so it would be AC.
So tools like eletric motors, drills, mixers, etc etc, use specifically Alternate Current or Direct Current? 
And what current do speakers need to work?
 A: You are asking about a very broad topic - let me just give a few pointers:
There are AC motors and DC motors. In principle, an AC motor may not need brushes if you tune it so the frequency of excitation matches the rotation frequency. In general, people will build these in a way that there are multiple poles on the windings - this provides more consistent torque (and guarantees a particular direction of rotation) where a "simple" coil and magnet may end up rotating either clockwise or counter clockwise.
It is possible to make a "brushless DC motor" by using an electrical circuit that reverses the direction of current through the coils; or you can uses brushes (so that when the coil reaches a certain position, the current is interrupted and current starts to flow in the next winding).
In the end, the concept for a motor is always this: turn on a current that causes attraction between two magnets (coil and permanent, or coil and coil); as the motor turns, you change which coil is energized, and what direction the current takes, to continue to generate torque in the same direction. Whether you need AC or DC for this depends in part on the operating characteristics of your motor (do you need variable speed/torque? How important is efficiency? Is your supply voltage AC or DC?). 
As for speakers: they operate on AC (because they translate variations in current into variations in air pressure, which you perceive as sound). But typically you need an amplifier to take the weak electrical signal (for example, from a microphone) into the strong currents for the speaker - and that amplifier usually operate on DC (and converts the DC supply into an AC current - just like the electronic commutator of the brushless DC motor turns DC into AC).
A tech note on the "AC or DC" choice can be obtained at http://www.dartcontrols.com/2013/09/ac-and-dc-motor-drives-is-one-the-clear-choice/
