The abbreviation "BBS" (or "bbs") stands for "baby's butt smooth" (or "baby-butt smooth," "baby's bottom smooth," and so on). It refers to a shave that leaves the skin of the face "as smooth as a baby's butt." This shave is considered the best possible shave. Although a shave might technically leave the face quite smooth but also irritate or cut the face, a "BBS" shave normally implies not only superlative smoothness but also a comfortable and successful shave.
A related set of three-character abbreviations is "WTG," "XTG," and "ATG" ("with the grain," "across the grain," and "against the grain," respectively). Shavers will combine more then one kind of "pass" in order to accomplish a BBS.