Pithing Needle

[Shackles]

So let me get this straight, and please correct me if I'm wrong, but abilities that cost purely mana will work and no others. Is that correct?

[Remillo]

Pithing Needle refers to Mana Abilities.  You're making a (normal) confusion between a Mana Ability and an Activated Ability that only costs mana. 

Mana Abilities are abilities that:
1) Don't target anything
2) Could add mana to a player's mana pool when it resolves
3) Isn't a loyalty ability (i.e. {Koth of the Hammer}.

That's the definition of a Mana ability.  Think {Llanowar Elves} or {Elvish Mystic}.  They simply add mana and do nothing else, so they wont be stopped by Pithing Needle.

Any other ability with a cost (a Mana Cost, Tapping, Sacrificing, etc., followed by a Colon) is a normal activated ability that WILL be stopped by the Needle.  Also, note, that while the abilities of {Deathrite Shaman} and {Arbor Elf} are often treated as Mana Abilities, they aren't.  So the Needle will shut them down.

[Mr_Fahrenheit]

Mana abilities do not use the stack. They just add mana to your mana pool in the same way as a basic land becoming tapped adds mana to your mana pool. Which is why Arbor elf and deathrite shamans abilities are not mana abilities, like Remillo said.