There are many types of external structures, some ornamental or architectural, some with a purpose or use, and some that are just nice to look at:
Bird tables and bird boxes are covered in the wildlife / birds section.
Bird houses range from small and simple constructions for a single breeding pair, to large architectural structures designed to house many birds.
Traditionally, a dovecote was a building for raising birds for food and there are many historical examples of them across the UK. In medieval Europe, the possession of a dovecote was a symbol of status and power and was consequently regulated by law. Many ancient manors in France and the United Kingdom have a dovecote. This is a photo of Dunster dovecote.
Modern dovecotes are much smaller in size and primarily used as architectural features or for conservation.
Modern gazebos tend to be wooden structures and often feature a thatched roof.
This design by the Breeze House is one we particularly like.
A pagoda is the general term in the English language for a tiered tower with multiple eaves but, in the context of a garden it usually refers to a small wooden structure with a roof.
A pavillion is defined as 'a light, sometimes ornamental roofed structure, used for amusement or shelter'.We really like this example which has glass walls.
A pergola refers to a (typically) wooden structure with an open roof, formed from poles or beams.
An arched pergola.