Impeachment is analogous to indictment in regular court proceedings; trial by the other house is analogous to the trial before judge and jury in regular courts. Typically, the lower house of the legislature impeaches the official and the upper house conducts the trial.
At the federal level of government, agencies such as the U.S. Department of Justice's Civil Rights Division, the U.S. Department of Education's Office for Civil Rights, and the Equal Employment Opportunity Commission (EEOC) investigate claims of civil rights violations and discrimination.
Just like the federal court system, most state courts have two separate court systems, one for criminal cases and another for civil cases. In our system, criminal courts are for cases in which someone is accused of a crime, and civil courts are for when individuals or corporations have a legal disagreement with each other that can not be resolved on its own.