A Breezeway is normally a connecting structure between two buildings, typically a house and a garage or barn or other "detached" structure. Breezeways originally were open on two sides to allow breezes to blow through.
The breezeway probably got its name from the fact that early breezeways were wide-open on two sides, allowing breezes to blow through freely. Whether the concept originated in hot climates (prior to air-conditioning) or elsewhere is not clear, but it is clear that the breezeway has taken on a variety of changes over time.
Today's breezeway is often a closed in area, whether just with screens or complete with walls and windows and even heat in some cases. Some homeowners even create lovely sitting areas or other functional uses for modern breezeways. Care for a dip in the breezeway hot-tub, anyone?