Chains date at least back to 2500 BC, and we can give credit to the Ancient Egyptians for threading links of gold and silver together - much like we thank them for hammering out sheets of gold and rolling it into early wire. Gold was plentiful in Classical Greece, and the Greeks created 3-dimensional pendants to wear on the chains. Around the 8th century BC, "strap chain" was developed, which is strands of chain that are connected to lie side-by-side, creating a fabric-like flexible band.
By 300 BC, the elite upper-class was wearing chains that contained gemstones in the links. Leonardo da Vinci improved on chain design during the Renaissance, a period when personal ornamentation was starting to become a must. In medieval times, the brooch was the most common piece of jewelry; in the Renaissance, this shifted to a pendant worn on a long gold chain.
For the upper classes, these lavish pendants often illustrated Biblical scenes, or emblems representing the names of saints or loved ones. More functional pendants have been found, as well, such as pendants containing toothpicks!
Fast-forward to the 18th century, and chain's use has remained much the same, with an added task: the muff, a lady's roll of fur to place her hands in to keep them toasty, often had a chain that went around the lady's neck. Called a muff chain, this is also seen in children's mittens!
As time passed, technology progressed to automate and standardize the making of chain, making steps like soldering the links (which often caused the entire link to become fused and unmovable) predictable and quick. There's a video at the bottom of this post showing just that!