Action point allocation is based on the AG stat. Therefore, a character that wishes to keep his enemy on its toes, or claws, needs to have a high Agility in order to have multiple combat moves. It is a critical stat for anyone interested in the more mobile and visceral skills, such as unarmed and small guns. Other skills rely on it, but these have higher values in AG investment. In Fallout Tactics, the Continuous Turn-Based (CTB) style required real-time generation of action points that was based on the character's AG stat.
Agility determines the number of Action points available for use in V.A.T.S. (base AP determined by 65 + Agility*2, and can be raised by perks and some items). Expert FPS players may wish to skip the use of V.A.T.S., but you should be familiar with the V.A.T.S. mechanics before you decide to do this, because V.A.T.S. not only increases your reaction speed and allows for more precision shooting, but also greatly reduces damage taken while it is active.
Contrary to Agility's description in game, and what the Fallout: New Vegas Official Game Guide states, Agility does not affect run speed. This can easily be proven in game with a stopwatch, measuring the time it takes to travel a given distance with a 1 Agility and a 10 Agility character (the same time).
If the player's Agility stat is the most extreme (highest or lowest), Doc Mitchell will either say "Most patients don't get out of bed after being shot and then move like they was in perfect control. You're unusual, I'll say that." (for highest) or "Don't have all of your coordination back yet, it looks like. You should think about doing some rehab." (For lowest)