Aspirin works by blocking the cyclooxygenase enzyme COX-1.  It acts as an irreversible inhibitor of the enzyme by adding an acetyl group to the active site.  This is an irreversible reaction, meaning that it cannot but undone or reversed.  COX-1 is an enzyme that is responsible for the synthesis of a group of hormones called prostaglandins.  The prostaglandins are responsible for inflammation (swelling) and the transmission of pain responses to the brain.  So by blocking the transmission of pain responses, aspirin helps reduce pain.  And by blocking inflammation, aspirin helps reduce swelling and localized pain.