Hello, and welcome to the forum!
A Layperson is a person who follows a particular calling without being ordained as a monk, priest, or nun.
An ordinary person, like you and me, who chooses to devote themselves to practice anyway.
Could be married, could be single, any age....
Many monasteries offer retreat periods for laypeople.
During these times they offer teaching of the Dhamma, and sessions of meditation.
You live the time according to their rules, generally waking when they awaken, following the day as any monk, might, and learning about monastic living without actually being ordained.
Now, as to what happens, when, and how, well, that would depend on where you are, what monastery is closest to you, which tradition they belong to, (Theravada? Mahayana which 'houses' Tibetan, Zen and Pure Land....) and whether they indeed offer such retreats to the public.
This is just an example, but in the South of England, you have the Amaravati Monastery.
This is their website:http://www.amaravati.org/
Other monasteries probably have similar websites about themselves, with all kinds of information and contact details.