I am a beginner in alpine skiing. It means I can ski parallel and now am trying to increase my speed plus avoiding awkward mistakes in turnings. I want to begin skiing in natural slops and off-piste. Since now I used narrow and very light skis in pistes and now need to choose appropriate skis for off-pistes. Can you describe differences between the equipments which I have to choose for these two different reasons if there is any?
When Off-piste, you will be skiing in deeper snow. For this purpose, it is helpful to have slightly wider skis. But to be honest, it mostly depends on your skill. Skiing in deep snow is different then on-piste. Very different.
My first times off-piste I went with a trainer, and the perfectly normal skis I use on-piste - which is what I recommend you to do as well for off-piste.
Be warned - you will spend much time digging yourself out of the snow and trying to get up - which is a much greater challenge when the ground keeps falling away from you (as soon as you push on the snow, it gives away, so you have to use your skis as leverage to get up comfortably).
The different ski equipment aims to enhance the fun-factor of off-piste skiing, but it does not replace the different skill needed to ski off-piste.
Good luck to you - but if you manage, you'll have a heck of a good time :D
If you're set on doing off-piste skiing and are prepared to invest a little into it, I strongly suggest getting a pair of "Big-mountain" skis. These are of the wider variant with bigger turning circle and are stiffer than "normal" skis.
As Rafael said, the equipment is only used to enhance the different factors, your skill is the most important factor.
Ps. if your set on going down natural slopes, get a good helmet, a shovel and some kind of Recco transmitter, for your own saftey.
Big mountain skis or offpiste have done a massive development in last 4 - 5 years and are significantly different from the regular slope or carving skis.
Main differentiating factors are
I personally would recommend to start with something like freestyle crossover skis, which are 50/50 piste/offpiste, they are fun on the slopes and also can be decently rode in forests and not too deep. Big fat ones are actually veery fun, but also quite challenging to ride.
Btw. they actually are significantly softer than the carver or piste skis, as you don't need that much control, which is the case while riding on the tracks, and also the weight considerations, so riding them on the piste is actually challenging as they tend to start vibrate at high speeds (going over 80 - 90 is pretty "interesting").
Good recommendations in addition to getting the shovel and some recco jackets, is actually take an avalanche course, it's going to be the decisive thing when issues happen. And going riding alone is always a bad idea.