Amberites have several powers which make them extremely powerful.
Their main ability shared across them, is their ability to walk between "Shadows" which are alternate universes created by the "Pattern". Amber itself is part of the "Pattern":
"We led our troops and saw them die, but of Shadow I have this to say: there is Shadow and there is Substance, and this is the root of all things. Of Substance, there is only Amber, the real city, upon the real Earth, which contains everything. Of Shadow, there is an infinitude of things. Every possibility exists somewhere as a Shadow of the real. Amber, by its very existence, has cast such in all directions. And what may one say of it beyond? Shadow extends from Amber to Chaos, and all things are possible within it. There are only three ways of traversing it, and each of them is difficult.
If one is a prince or princess of the blood, then one may walk, crossing through Shadows, forcing one‘s environment to change as one passes, until it is finally in precisely the shape one desires it, and there stop. The Shadow world is then one‘s own, save for family intrusions, to do with as one would. In such a place had I dwelled for centuries.
The second means is the cards, cast by Dworkin, Master of the Line, who had created them in our image, to facilitate communications between members of the royal family. He was the ancient artist to whom space and perspective meant nothing. He had made up the family Trumps, which permitted the willer to touch his brethren wherever they might be. I had a feeling that these had not been used in full accord with their author‘s intention.
The third was the Pattern, also drawn by Dworkin, which could only be walked by a member of our family. It initiated the walker into the system of the cards, as it were, and at its ending gave its walker the power to stride across Shadows.
The cards and the Pattern made for instant transport from Substance through Shadow. The other way, walking, was harder." - Chronicles of Amber, Nine Princes in Amber
They can even "pause" these Shadows and manipulate them in certain ways, Corwin himself did an unheard of fear, creating an entirely new Shadow.
For one, Amberites are very strong, Corwin and his brother Random was able to lift a car up and walk with it:
"'Well, whatever', I told him, 'let's get back onto the road and get moving, if we can.' The rear wheels were sunken up to their hubcaps, and as I stared at them, trying to decide the best way to go about things, Random called out, 'Okay, I've got the front bumper. You take the rear and we'll carry it back to the road—and we'd better deposit it in the left lane.' He wasn't kidding.
He'd said something about lesser gravitation, but I didn't feel that light. I knew I was strong, but I had my doubts about being able to raise the rear end of a Mercedes. But on the other hand, I had to try, since he seemed to expect it of me, and I couldn't tip him off as to any gaps in my memory.
So I stooped, squatted, grasped, and started to straighten my legs. With a sucking sound, the rear wheels freed themselves from the moist earth. I was holding my end of the car about two feet above the ground! It was heavy—damn! it was heavy!—but I could do it! With each step that I took, I sank about six inches into the ground. But I was carrying it. And Random was doing the same with his end. We set it down on the roadway, with a slight jouncing of springs." - Chronicles of Amber, Nine Princes in Amber
There are also ridiculous feats like, when Corwin was teamed up with Bleys, they single-handedly walked up a mountain slaughtering thousands of armed soldiers.
But in a direct fight, the one considered the best is definitely Benedict, he can beat people hundreds or even thousands of times faster then him and is easily one of the most skilled beings in all of fiction:
"Excuse me, please. You do not understand. You do not really understand who it was we talked with in the tent that night. He may have seemed an ordinary man to you – a handicapped one, at that. But this is not so. I fear Benedict. He is the Master of Arms for Amber. Can you conceive of a millennium? A thousand years? Several of them? Can you understand a man who, for almost every day of a lifetime like that, has spent some time dwelling with weapons, tactics, strategy? All that there is of military science thunders in his head. He has often journeyed from shadow to shadow, witnessing variation after variation on the same battle, with but slightly altered circumstances, in order to test his theories of warfare. He has commanded armies so vast that you could watch them march by day after day and see no end to the columns. Although he is inconvenienced by the loss of his arm, I would not wish to fight with him either with weapons or barehanded. It is fortunate that he has no designs upon the throne, or he would be occupying it right now. If he were, I believe that I would give up at this moment and pay him homage. I fear Benedict." - Chronicles of Amber, The Guns of Avalon
Merlin, Corwin's son, introduces Ghostwheel, a sentient, multi-dimensional quantum supercomputer he built in a Shadow where the laws of physics allowed it.
Ghostwheel can process and search infinite Shadows simultaneously, manipulate Trump dimensions, and operate at a vastly different scale than standard magical items:
“I designed and built a piece of data-processing equipment in a shadow where no ordinary computer could function, ” I replied, “because I used different materials, a radically different design, a different power source. I also chose a place where different physical laws apply, so that it could operate along different lines. I was then able to write programs for it which would not have operated on the shadow Earth where I’d been living. In doing so, I believe that I created a unique artifact. I called it the Ghostwheel because of certain aspects of its appearance.” - Chronicles of Amber, Trumps of Doom
There's the Jewel of Judgement, or Eye l of the Serpent, which has high-tier reality-warping powers, and scales to at least universal:
“The Jewel of Judgment and the Eye of Chaos or Eye of the Serpent are different names for the same stone?” I said.
“Yes, ” Dworkin replied.
“What happens if the Serpent gets its eye back?” I inquired.
“The universe will probably come to an end.” - Chronicles of Amber, Knights of Shadow