What is the Ash-Greninja form of Greninja? This new form of Greninja is circulating around. I am not following the anime or the movies so please try to be gentle if it was obvious.

What exactly is this Greninja? If it's a mega-evolution, how did it get there without a stone? How exactly does his powers/abilities/skills vary from other normal Greninja?

I know the physical difference, that it starts to resemble Ash a bit. Easily noticeable is the red middle part on his head instead of the body blue.

This is the normal Greninja: 



And this is Ash's:
Basically, what happens is that the bond between Ash and Greninja is so strong that Greninja took the new form as Ash-Greninja where Greninja takes some features of Ash (as you mentioned) and it is not a mega evolution. Some theorists say that it might be synchro evolution from the pokemon spin off Pokemon Reburst. By turning into this form, Greninja's power greatly increases and as you can see on its back, there is a giant shuriken which is a replacement for water shuriken (the move). The Ash-Greninja form can only last for a short period of time before Ash and Greninja gets tired and faint, during the form ash can also feel the pain that Greninja is feeling.

The official website explanation is "Ash's Greninja takes on a mysterious new appearance in the latest season! Ash-Greninja is the form that Greninja takes when the bond between it and Ash is raised to the limit. The strength of their bond changes Greninja's appearance, and it takes on the characteristic look of Ash's attire. This phenomenon is also said to have happened just once several hundred years ago in the Kalos region, but it remains shrouded in mystery."

The anime explanation is saying pretty much the same thing. When it first happened in the ninja village, the anime explained that because the bond between is great, he has taken a new form. The first appearance looked something like this.



during this stage, Stallone did a lot of testing where he discovered that this transformation will take place only when Greninja and Ash have mentally synchronized. Then later in episode 25, while versing the kalos champion, Ash-Greninja got the full form and it looked like this.



it was then explained that this was when the bond became maximum level where they are completely synchronized. And then it goes through the phases of difficulty between Ash Greninja and stuff.

Some power ups include faster, stronger, giant water shuriken, using water ninja knife things instead of the normal iyarigiri move. And in the episode 35, water shuriken became huge (kind of like rasen shuriken), it looked like this



Everything about Ash-Greninja is explained in the anime, it also shows Stallone testing how it works and the road to master this form.

I highly recommend Pokemon XY &amp; Z. The anime has improved a lot, Ash is no longer the immature young kid, he seems (mentally) more as a teen now and yeah, I definitely think you should give it a try.