Ok, so I play Shaman of the Pack (I'm using an elf deck) whilst having a few elves (6 or so), and my opponent takes the damage. His creatures are tapped out, and I have a 5/4 elf with an artifact equipped that gives +1/+1 for each creature that shares a type with it, and I begin to declare attackers. At this time, he decides to counter Shaman of the Pack. Is this a legal move or is he in the wrong?
If he has said that he takes the life loss from Shaman of the Pack then the spell has already resolved, entered the battlefield, the enter-the-battlefield-trigger has been put on the stack and resolved. It is then way too late to decide to counter the Shaman.
If he had not yet acknowledged the life loss from the Shaman verbally or by obvious gesture, then he still retains priority and you cannot decide to move to combat.
The important thing is that he acknowledged the loss of life.
It is too late for your opponent to counter Shaman.
You mentioned that your opponent took the damage from Shaman's "enters the battlefield" triggered ability. Your opponent cannot take damage from an "enters the battlefield" trigger and then decide to back the game up to counter the creature that entered.
The actual confusion seems to be about what can actually be countered, which the rules clear up in an unusually short manner:
701.5a. To counter a spell or ability means to cancel it, removing it from the stack. It doesn’t resolve and none of its effects occur. A countered spell is put into its owner’s graveyard.
This requires the spell to be actually on the stack in the first place, as indicated by Counterspells targetting spells. Once a creature spell resolves and as a result enters the battlefield, it can't be countered anymore or targeted at all by Counterspells.
Your opponent seems to be under the impression that a counterspell is about the same as a destruction spell such as Murder, which is not the case.
As mentioned in the other answers as well, the fact that your opponent acknowledged the life loss from an ETB ability means they agreed to advance the game to a point where that creature had already left the stack and entered the battlefield, making this situation particularily easy to resolve - your opponent is not able to counter your creature at this point anymore.