A docket is an official court record of everything that has happened and everything that has been filed with a particular court during the course of litigating a particular case. Each case heard before a court has its own docket. The docket will list the parties and lawyers in the case and basic administrative information about the case. Most importantly, it will contain a list by date of everything that has happened in the case. In a district court docket, for example, the first item on the docket will generally be the date the complaint was filed. In a Supreme Court docket, the first item will be the date the Petition for a writ of certiorari was filed with the Court. The docket will also list when the parties filed briefs and other significant events. For example, take a look at the docket for the case challenging President Obama's Patient Protection and Affordable Care Act.
With respect to the US Supreme Court, the following distinctions can be made:
A Petition for Certiorari is a document that the losing party in the appellate court (e.g. Ninth Circuit Court of Appeals) submits to the Supreme Court, asking the Court to review the appellate court's decision.
A Brief is a document a party submits to a court that attempts to persuade a court to agree with that party's position on the issue before the court. In Supreme Court cases, Merit Briefs are submitted by the parties after the Court has granted cert. Briefs are also submitted by "amicus curiae" (latin for "friend of the court"), which are parties who are not actually involved directly in the litigation, but have a strong interest in the outcome of the case.
Memorandum or Memo is the formal term for a "brief." Memo or Memorandum are also terms that we use to describe many different documents in the legal profession, such as an inter-office memorandum. A Cert Memo, specifically, is a document that a law clerk writes for a Supreme Court Justice, analyzing the petitions for certiorari and making a recommendation to the Justice as to whether the Court should hear the case. Cert Memos are confidential documents that are not often published; as such, they can be difficult to find (and may not be available for every case).
A Court Opinion is a lengthy document in which the Court explains its reasoning for ruling the way it did in a particular case. We often loosely use the term "case" to refer to Court Opinions (especially in law school). However, the term "case" also means a dispute between two (or more) parties that is being litigated before the Court, and includes everything that happens in the course of litigation between those parties (including opinions, briefs, hearings, etc.).
A Court Order is a ruling from the Court, affirming or overturning a ruling from the lower court. An order can be very short in length, but is often accompanied by a longer opinion, explaining the reasoning behind the order.
Court Rules are the rules that govern the procedure of the Court, for example, when briefs must be filed and how long a party has to respond to the opposing party's brief. The Supreme Court publishes its Rules of Court on its official website as a searchable pdf.