The Anchor itself has been around since the time of the Sumerians in 2000 - 2500 BC. It is one of oldest symbols used in tattoos, with the first images being used during the early days of Christianity at a time when Christians were persecuted for their beliefs.
Many circumvented persecution and disguised their affiliations by wearing an anchor, which not only masked the symbol of the Cross, but also represented Jesus' connection with the sea (several of his disciples were fishermen and he was also known to perform numerous miracles that involved marine elements). Indeed, the Anchor is a strong Christian symbol and is intricately woven throughout the Bible. It is also used as a representation of St. Clement of Rome, who was executed at sea by being tied to an anchor and thrown overboard.
According to some scholars, the Anchor also symbolizes the connection of Feminine and Masculine powers, with the crescent (the bottom half) of the anchor being the female element and the rod representing the male phallus. This ideology was commonly found in ancient Egypt, since the anchor is akin to the Ankh, the Egyptian symbol for Life.
It wasn't recognized as a proud calling card of sailors and servicemen until around the 1500s when it was adopted by the British Navy after a decisive victory over the Spanish naval fleet. In the early 1900s, Anchor tattoos were used to identify several generations of servicemen within a family, but the modernization of warfare has resulted in fewer family members enlisting in the military. As such, it is growing increasingly uncommon to see a grandfather, a father, and a son within the same family boasting their tats.
Today, the soaring popularity of anchor tattoos outside the Service is due in part to celebrities, who proudly display their works of art for all to see.