Bias is, in general, simply prejudice. It is an existing attitude that causes a person to look at a given circumstance in a way that is not objective and fair. People can be biased for many different reasons.
Typically, when we talk about bias we are talking about bias that is based on something like race, sex, or sexual orientation. There are other kinds of bias, but these are the sorts that are most often seen as important for our society. An example of bias, then, would be the attitude that men and women are fundamentally different and should be treated in different ways. A person who holds such biases might believe that women are inherently not cut out for jobs that require toughness. For example, a company might decline to promote a woman to a job that requires hard-nosed negotiations because they feel that women are not capable of being tough in such situations.
The major implication of bias is that it leads to unfair treatment of individuals. It also weakens our democratic society. When we exhibit bias, we treat people differently just because of what group (what sex, what race, etc.) they belong to. This is unfair and it is also bad for our society because it erodes the idea that all people should be treated equally and should be judged on their individual attributes.