The pilot chute acts as an anchor in the air, while the jumper continues to fall. As the two separate, the bag in which the canopy is folded is pulled from the container. The parachute's suspension lines, carefully stowed on the outside of the bag, are drawn out until they are fully extended. The bag is then pulled open and the canopy comes out. It immediately begins to inflate as the cells fill with air.
Inflation is slowed by the slider which prevents the canopy from expanding too fast.
It usually takes three to five seconds from deployment of the pilot chute to full inflation of the canopy. Over the years, parachute design has been refined to a remarkable degree. In fact, square parachutes have no known inherent design malfunctions. Theoretically, given proper packing, a stable deployment, and barring material flaws, a square parachute will never malfunction. However, we don't live in a perfect world, and malfunctions are common enough that no sensible person would intentionally jump without a reserve.