Judicial activism differs from judicial restraint in that judicial activism is when rulings are suspected of being based on political or personal considerations rather than on established law. While judicial restraint is when the courts deffer to the judgement of either congress or the executive branch.
A judge could be said to have ruled in an activist manner when the legal decision results in a check on power to both the legislative and executive branch, challenging that they have exceeded their authority such as legislation being ruled unconstitutional by the judicial branch.
We can tell if a judge has practiced judicial restraint when the ruling official defers judgement to those established by the executive and legislative branch of government, succeeding the jurisdiction of the matter over to ruling of another branch of government.
An appellate court is a court of second instance or any court of law that is empowered to hear an appeal of a trial court or other lower tribunal where as a district court is the general trial courts where both criminal and civil litigation is initially filed.The main difference in the two lies in their judicial jurisdictions where as cases are originally decided in the district courts where the decisions are then subject to appeal to the appellate courts after judgement.
The Supreme Court differs from the appellate court in that the Supreme Court was established as the highest federal court in the united states pursuance to Article III of the Constitution. This gives the Supreme Court ultimate appellate jurisdiction over all federal courts including the appellate court.