Instant casting means that there is no delay between the key press and the execution of the ability. Guards, movement skills and Warcries with the Battle Cry passive skill are a few examples of instant skills.
Instant skills can be used in the middle of executing a non-instant skill, or while moving. This still requires you to pay their cost and they can’t be activated while you can’t use skills, e.g. when frozen or stunned.
Some instant skills, like Frostblink or Vortex, will be used repeatedly as long as their bound key is held down, while others, like most reservation skills, will activate only once.
All instant skills have a cooldown.
Previously, we used "instant" to refer to skills with a cast time of zero. These skills took no time to do, but you still had to actually do them – this meant you couldn’t do them while doing other things, because the character can only have a single current action. So you have to finish your current action/stop moving, and then you could do the instant thing (taking no time), then go on to something else (or start moving again). So while skills were instant in that they themselves took no time to execute, because the character can only do one thing at once, there was generally some small delay becasue you had to stop/finish your other actions first, and start a new one after.
Now, "instant" for skills also means that you can "use" them without them ever being your current action – you can use the skill without that being the thing your character is doing – so you can use an instant skill in the middle of executing a non-instant skill, or while moving.
The implementation borrows from triggered skills, but unlike triggered skills, instant skills are still something you use, meaning you need to pay any costs, and won’t be able to do them when you can’t use skills – like while you’re frozen or stunned, for example.
tl;dr: Your character no longer has to actively "do" an instant skill when you use it.