Some names require a lot of digging to discover their meaning or origin. Surnames derived from place names often fall into this class as they can reflect the entire history of a region. English place names come from every period of its history, with many having obscure Celtic or even pre-Celtic origins. Other surnames have very transparent meanings. Bergmann is one such name. The name consists of two parts berg and mann which in German translate to "mountain" and "man", respectively. As a compound, the term is used for "miner", possibly offering a clue to your ancestors' occupation. Berg is also a common element in many German place names (Königsberg, Nürnberg) and can easily be confused with burg "fortress" (discussed here). This confusion is particularly common for English speakers who do not distinguish between short vowels /i, u, e/ followed by /r/ and thus pronounce burg and berg the same way.
Besides German, these same elements are also common in other Germanic languages. Berg also means "mountain" in Dutch and Swedish, while Danish has bjerg. The name Bergman without double /n/ is also a common Swedish surname, as with the director Ingmar Bergman and the actress Ingrid Bergman.