As for how the program works, according to the site's Frequently Asked Questions section, "Akinator uses the program Limule published by Elokence.com. The algorithm we use is an original creation. How we created it is our little secret."
There doesn't appear to be much more information available about just how the program uses Limule to make the program work so effectively.
Yes, there is a name for these classes of algorithms known as the classification algorithms in the field of machine learning.
In the classification problem, the algorithm has features that can answer the questions.
Deciding which question should be asked next can be done in various ways - for example by trying to maximize the predicted (or mean) entropy from the next question.