What happens after a purchase order is issued?

After a purchase order is issued, the supplier accepts it (explicitly or by starting fulfilment) — and from that moment it's a binding agreement. Then: the supplier ships or delivers; your side receives the goods and checks quantities against the PO (this receiving step is where discrepancies get caught, or don't); the supplier's invoice arrives and gets matched against the PO and the receiving record — the 'three-way match'; and once they agree, the invoice is approved and paid per the terms.

Partial deliveries are normal: the PO stays open with remaining quantities until everything has arrived, which is why tracking qty ordered vs qty received per line matters.

The whole lifecycle is bookkeeping-by-hand misery in spreadsheets and email — and largely automatic in software that links POs to receiving and stock levels.