Bond yields refer to the rate of return or interest paid to the bondholder. It is the premium that investors are paid for holding on to government or corporate debt. Basically, a bond is a loan and they function pretty much in the same way a local bank charges borrowers for interest on the loans it gives out.
Traders pay attention to bond yields because they reflect investor confidence. If there is a weak demand for a bond, its yields rise to attract more buyers. On the other hand, lower bond yields typically imply that there is a high demand from investors, either because they are confident that they will get paid back at maturity or that they feel it is a safe place to hold their assets.