The most prominent features of the tesla coil are the secondary coil, the long cylindrical thing that sticks up, and he top capacitance, the thing that looks like the top of Robbie the Robot.
The secondary coil is a single layer of wire wound around a form such as a plastic pipe, and has around 1000 turns of smallish gauge magnet wire. The size ranges from 26 gauge, up to 18 gauge, but these numbers are of course not cast in stone.
The top capacitance is usually in the shape of a sphere or toroid, with the toroid being the preferred shape. (A toroid is like a donut with a web across the hole in the middle. It is electrically connected to the secondary coil, and is physically connected to the form that the secondary is wound on. It does not move or rotate.
When it is operating the tesla coil puts out sparks. These sparks go to grounded objects, or just out into thin air. The sparks produce quite a racket and hearing protection is required for large coils, and is highly recommended for smaller coils too.