To expand a bit on what others have said:
Waffle Decider’s description of how the main deploys is substantially correct (depending on the precise system it’s called a throw-out pilot chute or a pull-out pilot chute; when I was active the TO was much more prevalent). So is Xema’s description of the reserve.
On any modern sport skydiving system both the main and reserve canopies are folded and packed into a deployment bag, which is literally a bag, open at one end. The canopy is folded up neatly and placed in the bag and then the suspension lines are carefully S-folded up against the bag opening, with each fold of line going through a big rubber band (called a “line stow”). Each rubber band is fastened to the d-bag and it goes through a grommet on the opening flap of the bag, so the folded suspension lines hold the bag shut.
When the pilot chute hits the air (about 120mph) it acts like an anchor - pilot chute stops, skydiver keeps going at ~120mph down. As they fall, the pilot chute pulls on a long nylon web called the bridle, which extracts a tiny curved metal pin holding the main container shut. Container opens and the d-bag is pulled out. The rubber bands make sure that the suspension lines pay out nice and neatly. As the lines reach full length they pull out of the last rubber bands and the d-bag “mouth” is now open. The folded canopy is pulled out of the bag and begins to expand.
There’s also a bit of material called a slider which controls opening shock and keeps the line groups separated.
The only major difference on the reserve canopy is that rather than an external hand-deployed pilot chute it’s got a big honking spring inside the pilot chute and it’s all packed shut into the reserve container. A handle on the harness has a steel cable with a pin on the end (that assembly is the ripcord), that pin goes through a little loop of material that holds all the reserve container flaps shut against the spring-loaded pilot chute.
Pull the ripcord and the pilot chute leaps free. It’s quite powerful - I’ve deployed mine in the rigging shop when I dropped it off for a regular repack (many riggers will have you do this so you can actually feel how hard it is to pull the emergency handle and so that you’re satisfied the last guy did his job) and it shot a good ways through the air - probably 15-20 feet.
However, and this is a big however, unless there’s sufficient airflow, all that happens is the pilot chute falls on the ground, the d-bag might fall out of the harness but that’s it. The canopy is not going to deploy - it is packed to stay in the bag so that it takes a certain amount of force to come out, otherwise it’d just “explode” out in a mess when used and overly-fast openings can be dangerous.
Ah, here’s a nice set of images showing the sequence: