Trader having stocks issue sell order through their brokers.
It is listed in the stock exchanges and aligned as a queue by lowest amount in the top.
Trader wanting to buy stocks issue buy order through their brokers and is listed. Highest bid is placed first.
When amount matches, a deal is made.
At the end of the day, once the markets closed, the deals are executed. The amounts/shares are transferred between the trader accounts.
Any mishaps/shortage is handled by stock exchange using caution deposit.