Could somebody please give a big picture description of what exactly is the object of study in the area of Algebraic Dynamics? Is it related to Dynamical Systems? If yes in what sense? Also, what is the main mathematical discipline underpinning Algebraic Dynamics? Is it algebraic geometry, differential geometry e.t.c.?
In algebraic dynamics one typically studies discrete dynamical systems on algebraic varieties. Such a system is given by a regular endomorphism $D: X \to X$ of a variety $X$.
The case over number fields is also called arithmetic dynamics...
That said, note also that Joseph Silverman writes in the introduction to The Arithmetic of Dynamical Systems, Springer 2007:
There is no firm line between arithmetic dynamics and algebraic dynamics, and indeed much of the material in this book is quite algebraic.
By the way, for a very gentle introduction to a little arithmetic dynamics, see A Glimpse of Arithmetic Dynamics by Grady and Poston. For additional introductory material, see Vivaldi's An introduction to arithmetic dynamics. Beyond that there is the article Current trends and open problems in arithmetic dynamics by Benedetto et al. More resources are available on Silverman's page for his abovementioned book. Also, Benjamin Dickman has recently kindly pointed out to me Benedetto's new book Dynamics in One Non-Archimedean Variable.
The Wiki article states that it is a combination of dynamical systems and number theory. I know it's a redirect, but WP's information on this point is probably reliable enough :)
(Are you checking here because you are not comfortable with WP info? It is a serious question which I'm curious about.)