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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 (sometimes abbreviated LBG, but LBG is also used to abbreviate "Lyman-break galaxy") is the brightest galaxy in a specified local region, e.g., within some maximum differential redshift for their radial distances and some maximum megaparsecs for their projected distances. 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.
Note that the abbreviation BCG is also used for blue compact galaxy.