What Does the Agility Stat Do

Final Fantasy Wiki

Speed (すばやさ, Subayasa?), also known as Dexterity, Agility, or Agl, is a recurring statistic from the series. It determines how often a character will attack. In games with Active Time Battle or Conditional Turn-Based Battle, it is used to determine the rate at which the ATB or Charge Time gauge rises. It can be augmented or reduced by Time Magic spells Haste or Slow.

Agility determines how often a character gets a turn. Agility can be increased by activating agility nodes on the Sphere Grid. Rikku has the best Agility on her path on the grid, Auron and Lulu are the worst, but eventually, any character can traverse any path. Agility Spheres can be farmed from the Fenrir on the Monster Arena to further boost Agility.

The benefit of raising agility caps off before reaching 255. The very first turn of battle may, from a technical point of view, come slightly quicker with higher agility, up as far as 250; though this is negated in the event of a Preemptive Strike or use of a First Strike weapon. Apart from this first turn, the effect of raising agility caps off at 170. This first turn effect is virtually negligible, therefore many players choose to stop raising Agility once they reach 170. It is worth noting that the Standard Sphere Grid in its default configuration already contains enough Agility nodes to reach 170.

Aeons also have their own stats separate from the characters'. They are mainly based on Yuna's stats, but also can be boosted manually and increase when the player fights more battles. Shiva is the fastest Aeon, and Ixion the slowest.

GameFAQs stat mechanics (take your time to read everything)

Update: Here comes the part you're interested in.

First, you must be introduced to what Tick Speed is:

From Terence's mechanics data:

"The CTB system works on a system of clock ticks, very similar to FFT. There are, of course, differences.... Each monster and party member have a counter. Every tick, this counter reduces by 1. At that point, anyone whose counter has dropped to 0 may take their next turn. It is not currently known exactly how the game resolves multiple characters reaching 0 at the same time. In addition to the counter, every battle object has a tick speed. This is an integer worked out from the character's Agility. Any action you take will be a multiple of this tick speed. When a turn is taken, it sets the counter to a value related to your tick speed and the ability used. Each ability has a Rank that designates exactly how fast the ability is. Lower ranked abilities take less time than higher ranked abilities, and the progression is linear: a Rank 8 ability will take eight times as long to recover from as opposed to a Rank 1 ability. The CTB window shows the next 16 turns that will be taken, in whatever order. It assumes that each character, when their next turn comes up, will select a Rank 3 move. Rank 3 is the default speed for any ability. Furthermore, the bar next to each character's 'turn' represents how many ticks away it is. With no bar showing, the turn is to be taken during the current tick. A full bright pink bar represents a turn 20 ticks away, and for each further color that is filled, another 20 ticks is added on. A full bar of the darkest color represents a turn that is 60 or more ticks away. Because of all this, there are very few formulas required to explain all this. We'll cover the most important one now: what the counter is set to when you take a turn. The exact value is: Counter = [Tick Speed * Rank * Haste Status] ...where Rank refers to the Rank of the ability used, and Haste Status is either 1/2 for Haste, 1 for Normal and 2 for Slow. Keep in mind that when under Haste status, your Counter will be rounded up if a fraction remains. This often results in abilities with odd-numbered Ranks used by characters with odd-numbered tick speeds being recovered from slightly faster than usual. Casting Haste and Slow also changes the current Counter. A successful casting of Haste on a character will halve their current Counter while rounding down, regardless of whether they were Slowed or not beforehand. The opposite happens with Slow; their Counter will be doubled regardless of whether they were in Haste or not. Also note that Dispelling either Haste or Slow will NOT change their Counter. Their Counter will continue to descend at the same high or low value until their next turn."

Agi to Agi = Tick speed
170 to 255 = 3
 98 to 169 = 4
 62 to  97 = 5
 44 to  61 = 6
 35 to  43 = 7
 29 to  34 = 8
 ......