Spheres

Regular Spheres; Power, Mana and Speed, are usually gained from enemies in most battles. Ability Spheres can be a bit harder to come by as only rarer enemies drop it, so stock up when you have the chance. You can also use Distillers or abilities on weapons to make enemies drop certain spheres, as well as using the Distill abilities from the sphere grid. Note that distilling a sphere from an enemy that already drops that sphere can cause it to drop double the usual amount.

Rarer spheres; such as Black/White Magic, Skill/Special and the Attribute related spheres, are usually found in treasure chests; some in the 6 Cloister of Trials, though some monsters may drop these spheres as well.

The very rare and useful spheres such as Warp/Teleport and Master are occasionaly given after boss fights, but usually require Bribing large amounts of gil to monsters to be aquired.
 

Heres a list of all the spheres and their effects:

Ability Sphere - Activates Ability, Skill, and Magic on the Sphere Grid
Accuracy Sphere - Turns an empty node into an accuracy node
Agility Sphere - Turns an empty node into an agility node
Attribute Sphere - Acquire an attribute already aquired by an ally
Blk Magic Sphere - Learn a black magic spell already learned by an ally
Clear Sphere - Clear a node from sphere grid
Defense Sphere - Turns an empty node into a defense node
Evasion Sphere - Turns an empty node into an evasion node
Fortune Sphere - Activates a Luck on Sphere Grid
Friend Sphere - Move one ally onto another ally's position
HP Sphere - Turns an empty node into a HP node
Lvl.1 Key Sphere - Unlock a Lvl.1 Lock
Lvl.2 Key Sphere - Unlock a Lvl.2 Lock
Lvl.3 Key Sphere - Unlock a Lvl.3 Lock
Lvl.4 Key Sphere - Unlock a Lvl.4 Lock
Luck Sphere - Turns an empty node into a Luck node
Magic Sphere - Turns an empty node into a Magic node
Magic Def Sphere - Turns an empty node into a Magic Defense node
Mana Sphere - Activates Magic, Magic Defense, or MP on the Sphere Grid
Master Sphere - Activate any ability learned by an ally
MP Sphere - Turns an empty node into an MP node
Power Sphere - Activates Strength, Defense, or HP on the Sphere Grid
Return Sphere - Returns the character back to any previously activated node
Skill Sphere - Learn a skill already learned by an ally
Special Sphere - Learn an ability already learned by an ally
Speed Sphere - Activates Agility, Accuracy, or Evasion on the Sphere Grid
Strength Sphere - Turns an empty node into a strength node
Teleport Sphere - Move to any already activated node on the sphere grid
Warp Sphere - Move to any node on the sphere grid
Wht Magic Sphere - Learn a white magic spell already learned by ally