It means the respiration through gills. Gills are special vascularised structures that are used as respiratory organs by aquatic arthropods and molluscs.
Branchial respiration is generally observed in fishes, aquatic arthropods, and in group Mollusca.
Most fish possess gills on either side of their head. Gills are the tissues that are made up of feathery structures called gill filaments providing a large surface area for the exchange of gases.
A large surface area is very crucial for the gaseous exchange in aquatic organisms as water contains very little amount of dissolved oxygen. Each filament comprises lamellae, which are discs supplied with capillaries.
Blood also moves in and out of the gills through these small blood vessels.
Though gills in the fish occupy only a small section of the body, the extensive respiratory surface produced by the filaments renders the whole organism with efficient gas exchange.