Cooldown is a game mechanic in Xenoblade Chronicles. It prevents the player from using one art several times in a row endlessly.
Once an art is used, it will be greyed out. This is a sign that the art is in cooldown. If you observe the art icon carefully, progressing upwards, the art will slowly start to turn a lighter shade of grey. When the lighter shade reaches the top of the art, the art has finished cooldown and can be used again.
The more powerful the art, the longer the cooldown generally takes, however using Art Points (APs) to upgrade arts will, as well as increasing the power or effect time, reduce the time needed for cooldown (measured in seconds). Although the way each art is improved upon leveling varies, all arts will at least have their cooldown times reduced upon leveling up.
Cooldown gauges are ignored during Chain Attacks, and all arts will become available upon starting the attack. Once the attack is over, each art's gauge will return to what it was prior to starting it.
Talent Arts, however, do not have a standard cooldown, and the player must manually fill the gauges for their Talent Arts. More info on this can be found on the page for the Talent Gauge.