After Cast Delay
After you use a skill, there is a certain amount of time where you can't use the same skill, or other skills. This is what is called Cast Delay.
There are technically three forms of Cast Delay: Animation Delay, After Cast Delay and Skill Delay.
Animation delay is the amount of time it takes your character to go through the motion of its cast (e.g. Heal), attack (e.g Meteor Assault), or throw animation (e.g. Acid Demonstration), which can always be reduced by higher ASPD or a technique called "Animation Delay Cancel" (*). You generally cannot initiate another skill until you have finished your initial animation for the previous one.
After Cast Dealy (on TalonRO, Cooldown)
After Cast Delay, on the other hand, is a fixed amount of time after which no additional skill can be directly initiated, which is indicated by a grey-colored pawnprint over all skills. Bragi and gears that reduce "after-cast" or "re-use" delay will shorten this time. Even at 100% reduction, this delay is hard-capped at about 300ms or 3 skills per second (officially and here; note that most eAthena servers do not set this delay by default and thus can spam skills much, much faster, especially high rates).
Skill Delay further prevents you from using a particular skill again, but you can use other skills. It's the pink-colored pawprint over an individual skill instead of grey over all of them, and is not reduced by anything. For instance, consider the Endure skill. You can't use it again, even if the effect runs out, for several seconds, but this would be disastrous if you couldn't use other skills in the interim. Only reuse of Endure is prevented for the full duration.
How After Cast Delay Works
The ability to follow one skill by another is limited by the highest of those three factors.
Jupitel Thunder for example has neither Skill Delay nor After Cast Delay and therefore is only limited by the Animation Delay of the cast animation the High Wizard has to go through. That means that increasing ASPD will increase over all DPS (granted, the Cast Time is low enough/instant), because it allows you to reuse the skill much more frequently.
That means generally, if you are able to reduce the After Cast Delay of any skill that has no Skill Delay, to a value smaller than its Animation Delay, ASPD or "Animation Delay Cancel" will help to reuse the skill more frequently.
(*) The effect of ASPD or "Animation Delay Cancel" greatly varies between skills. As a general rule of thumb, enemy-targeted offensive skills usually have ASPD-dependent animation delays, while ground-targeted and support skills usually have fixed animation delays, but there is no particular hard-and-fast rule for this. That means though, that if your skills Animation Delay is not really affected by ASPD, "Animation Delay Cancel" becomes more viable.
Some skills do lose their Animation Delay entirely when Instant Cast (150 dex, or 100% Cast Time reduction) is achieved. Examples are Acid Demonstration and Increase Agility. One would conclude that this means their reusability greatly increases, but thats weirdly enough not the case. The reusability for Acid Demonstration, with bragi, is best at 149 dex (with an animation) and not 150 dex.
The cap of 3 skills/s is reached at 184 ASPD. As an example, the reason why most Staff of Magi are carded with 3 Doppleganger Cards instead of any other combination that results in less than 190 ASPD is, that Ping plays a role in the ability to reuse skills. While someone with a Ping of 15 ms might be able to reach the 3 skill/s cap, someone with 200 ms might not, and thats where the higher ASPD helps in getting more skills out.