The gymnastics handstand is only one style of handstand, but probably the most popular today. It refers to a straight bodied handstand, as everything in gymnastics is about straight lines and looking good.
This video gives you a few ideas on why this is and how to train it.
I also cover the straight bodied handstand in my course Secrets of the Handstand, although I focus more on the curved handstand as it is an easier skill to learn. And once you have it you can then move onto the more advanced gymnastics style handstand. But some people would disagree with me, and just go for the straight handstand right from the get-go. It will take longer but it can be done.
The best drill in my opinion for getting the straight body position is to do a handstand against the wall, with your back facing away from the wall. From there have someone coach you into the straight position.