All Americans have a right to a fair trial when accused of crimes and facing charges. The entire legal system is based around the idea that people are to be presumed innocent unless they are found guilty in court, often by a jury of their peers. This jury is meant to be impartial and free from bias.
When picking the jury, some jurors may be excluded if bias is noted. This can happen even if the person insists that he or she isn't actually biased and will be fair and just during the case.
For example, if a police officer is involved in an assault case, a juror whose father and sister are police officers may be excluded from the jury. That person may claim he or she can still be objective, but the court may be worried that subconscious bias will work its way into the case since the juror grew up around police officers and has a close connection to them. They need a juror who does not have any connection -- good or bad -- to the police force.
The same can be done when jurors talk about their personal opinions and feelings. For example, before a drug case, a juror may mention that he or she doesn't think the War on Drugs goes far enough and that drug laws are too lenient. That juror could be excluded because he or she is biased against the very drugs laws and regulations that make the case possible, meaning the juror may unfairly be biased against the person accused of breaking said laws.
Remember, the right to a fair trial is one of your most important rights as a citizen. If you ever think it's being compromised, you need to know what legal options you have.
Source: FindLaw, "Criminal Trial Overview," accessed Dec. 23, 2016