As we previously explained, it is an ancient custom to use a ring for the wedding ceremony to do the Kiddushin or consecration of the wife (see here). According to some authorities (Tosafot) a ring was the normal means of binding a marriage, even in the time of the patriarchs as it could be interpreted from the story of Yehuda and Tamar (Bereshit 38:18).
Today the custom of the wedding ring or band is almost universal. However there are some communities that use a coin rather than a ring for marriage. For example, in some Syrian and Iraqi communities this is the prevalent custom (Ben Ish Chay). In the times of rabbi Sa'adia Gaon the tradition was to make the kiddushin with the cup over which the wine blessing was said. In other places, like Georgia (in Eurasia) it was the custom to place the ring in a cup.
One of the reasons why a ring is used is that a ring is worn all the time, as a permanent reminder of love and commitment. The rabbis compared the pact between God and Israel as a wedding. Every Jew, when he wears the Tefilin, winds the straps around his middle finger saying: "ve-erastikh li le'olam.... "., "I (=God) will betroth you for ever" etc., symbolizing that in the same way the wedding band represents the permanent union between husband and wife, the Tefilin is a symbol of the constant love and commitment between Israel and God.
A very original reason brought by Keter Shem Tob for using a ring is that giving a ring symbolizes giving over of authority. Pharaoh transfered his authority to Yosef and so did Achashverosh with Haman and later to Esther: the monarchs were empowering them and giving them authority over their kingdom. Similarly, "In giving his wife a ring, the husband is symbolically giving her authority over his household and everything else that is his. From that moment on, everything in their lives will be shared".
(Rabbi Aryeh Kaplan, "Made in Heaven").