A peristyle is a courtyard with a covered walkway all the way around it, with columns holding up the ceiling so you can see out into the garden. Peri means “around” and style means “column”, so a peristyle is a place with columns all the way around it. The walkway and columns were usually limestone. Richer people might use marble.
The earliest peristyle courtyards that we know of were in Greek houses, beginning around the Classical period. Peristyles were also common in wealthier Roman houses, all over the Roman Empire, beginning about the time of the Julio-Claudian Emperors. Many Roman people built their peristyle out of brick, covered with plaster and painted to look like marble – that was cheaper.
Because the weather in Greece and around the Mediterranean Sea was generally warm and sunny, people could work in the shade of the peristyle columns and have good natural light from the courtyard or garden in the middle. In ancient houses, the peristyle was a nice cool shady place to spin, or weave, or play with the baby, or break the ends off green beans.