It’s not just abilities Akatsuki rings have, but rather a will to fight.
A lot of members of the Akatsuki are missing limbs, and they wear these rings on their limited fingers in order to compensate for weaknesses or lack there-of. The symbolism here is that they’re missing something, and need this ring as artificial substitution until they can reclaim what’s been taken from them in some way. This obsession with claiming what has been denied them is part of their motivation for joining the organization in spite of its dishonorable origins – all for power or revenge against those who wronged them first.
Akatsuki understands that if it’s just about taking an opponent down then all you need is brute strength; so it chose something that’s more symbolic as a ring. It represents something that is worth keeping and protecting, and thus is willing to go out and fight for it. Hence its members will often take extreme measures to carry out their goals, so much so they’d be branded as terrorists for some of their actions even by civil standards.
Regardless of the Akatsuki member, they’re all united in their one-ness of purpose and the Ring is the only sign required to show that unity.
It’s not just abilities Akatsuki rings have, but rather a will to fight.
It reflects who they are as people and what they believe in and fight for above everything else – those values which made them join the organization in spite of its dishonorable origins.
Most members wear the rings on their pointing fingers, but there are some who wear them on another digit or even palms instead of two fingers – Madara and Zetsu always wore theirs on both index and middle finger and Kakuzu and Hidan had one on each hand. They’re black because they’re the Akatsuki rings no matter what they look like, but there are some who do have slightly different colors to represent who they are.
The one ring rule is not limited to just Akatsuki, but applies to everyone affiliated with them as well – this also includes Kabuto and Orochimaru during their brief membership in the group.