Not just any electricity will do. Specifically, it has to be moving current, and an alternating or AC current at that. The frequency of the current matches the sound that you want to come out of the speaker. Having multiple currents at different frequencies flowing at the same time is just fine, that's how a speaker can simultaneously reproduce a high note from a guitar and a low note from a bass at the same time.

The current flows through a coil that is part of an electromagnet. The voice coil, as it's called, converts electrical energy to magnetic energy. The frequency of the magnetic field matches that of the electric current flowing through the voice coil. So, we've just transformed an invisible alternating (vibrating, if you will) electrical current into an invisible alternating or vibrating magnetic field.

Now, if you place a permanent magnet in a magnetic field, there will be a force (a "push") acting on the magnet. If the magnetic field alternates, then the force on the permanent magnet will alternate, too. The magnet will want to move back and forth in resonance with the magnetic field.

The permanent magnet (or speaker magnet) is attached to the pointy end of the speaker cone. The cone acts like a very wide, thin, lightweight piston, pushing air back and forth. Because it is so thin, it has to be made in the shape of a shallow cone or else it would collapse when the speaker magnet pushes against it. All the speaker cone has to do is move the air back and forth when the speaker magnet pushes on it.

And because the speaker magnet pushes back and forth, the speaker cone moves back and forth in resonance with the force on the magnet, which is in resonance with the alternating magnetic field, which is in resonance with the electric current you fed into the voice coil. If you look at a bass speaker with the cover off, you can see it moving in and out when you use it to make sound.

Sound is just vibrating air. So when the speaker cone moves back and forth, it grabs a bit of air and moves it back and forth as well. That converts the mechanical energy of the speaker cone into sound energy. The air molecules affect the air molecules next to them, which affect the next closest molecules, and so on until the entire room is filled with sound.

And that's how electricity is converted into sound energy.