To swoon is to faint, due to lack of blood to the brain. Illness, fear, stress, and even happiness can cause people to swoon.
Swoon is an old-fashioned way to say "pass out": back in the day, ladies in corsets were always swooning at the slightest stress or smallest nod from a handsome man. Although this word technically means "to faint," today it has more of a positive connotation and isn't used so literally — people say they swoon when they encounter something so wonderful it makes them dizzy with joy. Chocolate cupcakes, the perfect dress, or a stellar home run could make you swoon.