Acquiring Anima is sort of a side quest involving several steps. Find the hidden location Baaj Temple with the airship. Run up the familiar path and dive into the water. Then prepare to finally defeat Geosgaeno, and swim toward the underwater temple entrance.
Geosgaeno: Geosgaeno attacks with Stone Punch and KO Punch, so equip Rikku, Tidus, and Wakka with armor bearing protection against petrification and death. Cast Hastega and Protect on the party, use everyone's Overdives.

After defeating Geosgaeno, swim north into the temple entrance. Enter the Chamber of the Fayth, where you'll find four Mega Phoenix items and a Megalixir. Approach the six statues on either side. There is one hidden item in each of Spira's six Cloisters of Trails. You should have five of these items at this point, and five of the statues in this room will activate as a result. Now you just need to get the last item from Zanarkand Dome.

Flay back to Zanarkand and enter the dome. There, a new puzzle solution is displayed on-screen. Step on the three white squares in the first room and the four in the larger chamber. This opens a panel in the first room revealing a Destruction Sphere. Place the Destruction Sphere into the slot to the right of the monitor in the larger chamber, and the screen will explode. Collect the Magistral Rod, and return to Baaj Temple.

The six statues will now resonate with the items you have collected, allowing you to enter the Chamber of the Fayth. Now this unbelievably power aeon is yours!