In this practice, the girl's father decides to conduct the Swayamvara of the daughter at an auspicious time and venue, and broadcasts the news of this to the outside world. Kings, typically used to send messengers to outside lands whereas commoners arranged to send the news across within the community.
On the appointed day and venue, a list of suitors arrive at the girl's abode and ask for her hand. The girl and her family get to choose among the suitors, sometimes through evaluating the completion of various tasks assigned. For example, in the Hindu epic Ramayana, Sita is married to Rama the only one strong enough to lift the Shiv Dhanush and string it.
Another famous Swayamvara is that of Draupadi of Mahabharata. Aspirants had to hit a roof-mounted revolving image of a fish with a bow and arrow by only looking at the reflection of the fish in the pond below the roof mount. Prince Arjuna, the third among the Pandavas, succeeds in hitting the fish. Draupadi disqualifies Karna from this contest as he is not a pure Kshatriya.
When the girl identifies the husband of her choice, she garlands him and a marriage ceremony is held immediately.
This practice seems to have been followed in the Western World too, at various times. In William Shakespeare's The Merchant of Venice, Portia chooses Bassanio through a test of his love for her over gold.