The answer in short? Almost infinite!
However, in Russian school, four major kinds are recognised.
The first and the second arabesques are universal.
The first arabesque is when a dancer stands sideways to the audience with his/her side to the audience, stand on the leg away from the audience with the other leg extended behind, with his/her supporting leg-side arm stretched in front and his/her other arm on his/her side pushed slightly backwards so it is not just on the side, both hands with their palms facing downwards. Both the dancer’s arm position and the leg position are open towards the audience.
The second arabesque is with the same leg position as the first arabesque, but with the audience side arm stretched in front with the right arm on his/her side, again pushed slightly backwards.
The third arabesque is when a dancer stands on his/her audience side leg, facing corner 2 or facing the audience side corner, to the audience’s left. The position of the arms are open again, i.e. his/her right hand towards the corner he/she is facing with his/her left arm stretched to his/her side pushed slightly backwards.
If you have heard of the third arabesque as something different, it might have been what Russian’s call Cechetti’s third. Cichetti’s third position usually refers to an arm position where both arms are extended forward in parallel but one arm (usually the arm towards the back of the stage) is held higher than the other.
The fourth arabesque is also with a dancer’s legs in crossed position as in the third arabesque but the audience side leg extended forward. In the fourth arabesque, a dancer’s shoulders are pulled across a lot further than in any other arabesques. This arabesque is half turned away from the audience, the head is turned towards the audience, emphasising its direction by the glance. This is no doubt the most difficult arabesque.
Apart from those arabesques, there are many variations of them with different arms. Even the first arabesque can be very different from role to role. For example, the first arabesque of the White Swan is very different from that of the Black Swan, which is again very different from that of Giselle depending on the angle of the arms and the inclination of the head.
In short, if a dancer is standing on one leg with the other extended behind, is it an arabesque! See just how many arabesques you can discover!