During launching the main thing is all steps of launching are done on time. that's why all say Rocket Science is very tough. The procedures for each launch are written carefully. A single-degree angle more inclination of Rocket can damage the trajectory that it follows. A countdown is a sequence of backward counting to indicate the time remaining before an event is scheduled to occur. The purpose of a countdown is to make sure all the engineers know exactly when liftoff should be, and how much time they have to make last-minute adjustments. The countdowns can start from 72 to 96 hours before launch time.
There is a long list of things that have to happen to launch a rocket after it gets to the pad. Fueling, engine gimbal tests, sensor checks, computer startups, etc, etc. There's a certain order these have to go in. A countdown provides the backbone for the planning of all this. Don't think of it so much as "time until launch," but rather "progress through the checklist." Sure, the launch does happen at T-0, but in some launches, when T-0 actually is can change if they hit a problem earlier in the countdown. Proceeding with the countdown depends on several factors, such as the proper launch window, weather that permits a safe launch, and the rocket and payload working properly.
In essence, normally when you hear "the countdown has been stopped" your gut feeling would be "something went really bad", but the reality is each countdown is stopped a few times normally - these pauses are planned. They allow for these last-minute adjustments that tests have shown to be required; they allow re-synchronizing the countdown in case some tasks took longer than expected, delaying others in consequence. They often overlap lengthy procedures like filling the fuel tanks, but some of them are pauses on their own. The countdown sequence is not written in stone - sure the launch must fit in the launch window (a couple hours) and there's very little deviation allowed during the final seconds, but the countdown a couple hours before the launch is more of guidelines "this should be done about now" than "T-16:36:58.07: flight captain uses a spoon to break the shell of the hard-boiled egg of his breakfast" type of lists.