A brightest cluster galaxy (or BCG) is the brightest galaxy in a galaxy cluster. A typical brightest cluster galaxy is a giant elliptical galaxy in the center of the cluster. They are generally thought to have grown through stripping stars off other galaxies or merging with them.
A locally brightest galaxy (or LBG, but LBG is also used to abbreviate "Lyman-break galaxy") is the brightest galaxy in a specified local region, e.g., specifying differential redshift for the radial distance and megaparsecs for the projected distance. The concept is used in selecting galaxies from large surveys, for example, to indicate the galaxies that are "primary" (brightest) within their dark matter halos.