A civil trial is when a person or entity sues somebody for one or more reasons. For example, a person can be civilly sued if they cause a car crash, and cause injuries. Civil lawsuits can be brought in both state and federal court, individuals, corporations, or governments can bring civil lawsuits against persons or entities for violating civil or constitutional rights. An example of this would be somebody suing somebody else for not fulfilling a contract.
A criminal trial is generally charged in a formal accusation called an "indictment". The government, on behalf of the people of The United States, will prosecute this case through the United States Attorney Office if this person committed a federal crime. State's Attorney Offices handle state crimes. An example of this could be somebody committing a violent crime that results in the deaths or harm of others. In instances like this, it is typically up to the government to bring charges against this person, not the victims.