Explain how the United States courts of appeals, or circuit courts, differ in their purpose from federal district courts?

The United States courts of appeals (circuit of courts) differ in their purpose from federal district courts in that courts of appeals function as a system to check the performance of federal district court judges.

Federal district courts have the right of working on cases which involve federal laws.

As regards the US courts of appeals, they are forums in which decisions made in federal courts, may be considered again. It doesn't mean that there will be another trial, it means that the courts of appeals may consider any possible mistakes and will be willing to check the performance of the federal cort judges.