Pointed Discussion

A classic. It protects you from damage, gives you a blocker, lets you use that creature's tap ability twice, ramps your mana with cards like Tolarian Academy, Sol Ring or just Tron. It protects you from counter-spells by tapping out your opponent's last island, you can put it on Scepter to get an Icy Manipulator on crack. It lets you use your artifact again and de-activates artifacts like Winter Orb or Howling Mine. There have been many variations of this card, many of which are a tad stronger. But Twiddle will always maintain its status as the first.

"Okay, Pact of Negation."
-next turn-
"good, i have just enough lands!"
"Twiddle, you lose...hehe..."

...and in response they tap their land for mana.

The best of the three arts on this card.

The art in fact shows a mage battling a Phyrexian. See Reckless Assault, a B/R card from Invasion, for similarity.

This card was pretty good back in the day, as it could untap cards like Mana Vault, Time Vault and Colossus of Sardia. Still decent today.

Yeah, jugglinguy is wrong, the mana source is faster than instant speed, so twiddling to counteract pact of negation doesn't work.
On the other hand, kitsunewarlock's idea about tapping their last island will work. Simply play the twiddle on an island during the Untap phase, or, if you just drew it, play it on your first main phase and cast the spell you want to save from countering on your second main phase. They can tap the island for mana but it will fade away between the phases.