The resources below provide great background and understanding of the United States Court System. If you are unfamiliar with the structure of the United States District Courts, Courts of Appeals and/or Supreme Court, it may be beneficial to first review the overviews set forth below:
The federal judicial system comprises a three-tiered hierarchical court structure. At the bottom tier are the district courts. District courts are the trial courts for the federal judicial system. There are 94 federal district courts, with at least one district court in every state. Trials conducted by district court judges are governed by the Federal Rules of Civil Procedure, Federal Rules of Criminal Procedure, and the Federal Rules of Evidence.
The middle tier comprises 13 appellate courts known as the United States of Appeals or the circuit courts. Appeals heard by appellate court judges are governed by the Federal Rules of Appellate Procedure. These courts include the United States Courts of Appeals for the District of Columbia Circuit, and the United States Court of Appeals for the Federal Circuit. Each of the 11 numbered circuit cours and the D.C. Circuit are composed of federal judicial districts within that circuit and hear appeals from those district courts.
In the federal system, the United States Supreme Court is the court of last resort and thus constitutes the top of the judicial branch's three-tier structure.
Federal appellate practice and procedure in a nutshell by 
Supreme Court and Appellate Advocacy: Mastering Oral Argument by US Appellate Courts
A. Geographic Jurisdiction: District of Delaware; District of New Jersey; Eastern, Middle and Western Districts of Pennsylvania; and District Court of the Virgin Islands.
B. Emphasis on the Standard of Review: The 3rd Circuit places high emphasis on the standard of review. Under the Federal Rules, the appellant's brief must provide a statement of the standard of review. The appellee may also include such a statement. Federal Rule of Appellate Procedure 28(a)(9)(B) permits identification of the standard of review within the discussion of the arguments. The 3rd Circuit requires that the statement be presented under a separate heading and placed before the discussion of the corresponding issue in the argument section of the brief. The 3rd Circuit has also repeatedly emphasized that the standard of review is often outcome-determinative of an appeal.
3rd Circuit Local Appellate Rule 28.1(b) identifies five different standards of review: abuse of discretion, clearly erroneous, plenary, substantial evidence, and the statutory standard of "review according to 5 U.S.C. 706." The court also recognizes the common law standard of plain error.
C. Motions: May be decided by a clerk, a single judge, or a panel, depending on the relief sought. The 3rd Circuit has a relatively liberal policy regarding motions for extensions of time with regard to the filing of briefs, in that such motions are generally granted.
D. Mediation: See Local Appellate Rule 33, et seq. Mediation program is mandatory in that if a case is referred to mediation, the litigants generally must participate. In the 3rd Circuit, briefing is not required until after it has been decided that a case will not be scheduled for mediation, or after the case has failed to settle through mediation. In the 3rd Circuit, senior circuit judges and senior district judges often serve as mediators, in addition to the full-time mediator employed by the court.
E. Electronic Filing: The 3rd Circuit requires that all documents be filed electronically through the CM/ECF system, with the exception of "original petitions such as a petition for writ of mandamus or petition for review of an agency order." There is an option with respect to the filing of the appendix. The court also requires 10 paper copies of briefs and fourt paper copies of appendices.
F. Briefs and Appendices: The 3rd Circuit has a few requirements for the content and format of briefs not presented in most other circuits. A Statement of the Standard of Review must be placed before the discussion of the issues in the argument section, under its own heading. The Statment of Issues section of the brief should cite to specific pages in the appendix or places in the proceedings at which each issue was raised, objected to, and ruled upon. The 3rd Circuit also requires that the appellant provide a statement of related cases, and that the Statement of Facts be supported by references to the record. For any legal proposition supported by citation anywhere in the brief, the brief must also cite any binding opposing authority of which counsel is aware. The 3rd Circuit does not receive a full record of the proceedings below; it relies on the docket entries and other materials provided in the appendices to the briefs. The appellant must file an appendix with the appellant's brief. The appellant must include in the appendix additional parts of the record designated by the appellee.
G. Oral Arguments: The 3rd Circuit hears oral arguments in 22.4% of cases, which is lower than the national average of 30.1%. The granting of oral agrument in the 3rd Circuit is unique in that the judges of the panels select the cases that will be set for oral agrument.
H. Decisions: The 3rd Circuit issues precedential and nonprecedential opinions. There is a 60 day aspirational time limit for authoring an opinion. After a second panel member approves a draft opinion, a 45 day aspirational time limit begins for a disproving third panel member to write a concurring or dissenting opinion.
I. Petitions for Rehearing: The 3rd Circuit follows the Federal Rules of Appellate Procedure regarding rehearings, with few additional features. The 3rd Circuit presumes that all petitions for rehearing request both a panel and en banc rehearing, unless otherwise explicitly stated. In addition to certain time notices for petitions and responses, the 3rd Circuit has a strong policy of avoiding intracircuit conflicts. No panel can overrule the precedential opinion of a previous panel. A decision of the 3rd Circuit panel can only be overruled by rehearing en banc. Still, petitions for rehearings en banc are not favored and will not ordinarily be granted where the panel's statement of the law is correct, and the controverted issue is one of application of the law to the facts of the case.
J. Award of Costs: The costs of reproducing and binding briefs and appendices is typically recoverable but subject to strict guidelines. Other costs, such as attorney's fees, are generally not recoverable absent specific statutory authority.
K. Imposition of Sanctions: The 3rd Circuit has traditionally been the most reluctant of the circuit courts to impose sanctions, but recently has become more aggressive in imposing sanctions for frivolous appeals under Federal Rule of Appellate Procedure 38. The 3rd Circuit typically awards attorney's fees incurred in defending the appeal, and declines to award double costs or other forms of punitive awards. The 3rd Circuit generally calculates the amount of the award itself, and usually imposes the award against the offending counsel rather than the party. The 3rd Circuit has not relied on 28 U.S.C. 1912 or 1927 to make an award for delay or excessive costs. The 3rd Circuit has established its own rules for disciplinary enforcement, which are available on its web site at http://www.ca3.uscourts.gov.
When state appellate rights have been exhausted and/or a federal case is moving to its next level of appeal, it is important to review the appropriate federal circuit court's rules and procedures. Below are links to the websites for the Federal Circuit Courts of Appeals. Great resources provided by each Court's website include, inter alia, Court Opinions, Local Court Rules/Procedures, Court's Guide/Guidelines (also Practitioner Handbooks) and Forms.