A marriage bann is usually a verbal announcement, typically in a church, that a couple intends to get married. In some cases, they may be published and publicly posted for people to read instead of being verbally announced but the banns are a chance for those who have knowledge why the couple should not be married to state that fact.
A marriage bond is a legal document guaranteeing that the person getting married has no legal impediment to marriage. Marriage bonds usually have a stated financial value that is only to be paid if the person for whom the bond was signed was not legally able to get married. The person getting married signs the bond and usually one or two bondsmen sign it as well. The bondsmen likely know the marrying party well enough to know that they have no legal challenges with getting married.
Banns and bonds serve a similar purpose, but they are different.
Wedding bands are something else entirely–the ring on the finger. The band at a wedding may also play music <grin>.