Matching of cases to federal judges

From The Federal Judicial Center's page on How the Federal Courts Are Organized FAQ:

How are cases assigned to judges? 
Each court with more than one judge must determine a procedure for assigning cases to judges. Most district and bankruptcy courts use random assignment, which helps to ensure a fair distribution of cases and also prevents "judge shopping," or parties’ attempts to have their cases heard by the judge who they believe will act most favorably. Other courts assign cases by rotation, subject matter, or geographic division of the court. In courts of appeals, cases are usually assigned by random means to three-judge panels.

Some other interesting Qs and As:

What is an Article III judge?
The U.S. Supreme Court, the federal courts of appeals and district courts, and the U.S. Court of International Trade are established under Article III of the Constitution. Justices and judges of these courts, known as Article III judges, exercise what Article III calls "the judicial power of the United States."

Are there judges in the federal courts other than Article III judges?
Bankruptcy judges and magistrate judges conduct some of the proceedings held in federal courts. Bankruptcy judges handle almost all bankruptcy matters, in bankruptcy courts that are technically included in the district courts but function as separate entities. Magistrate judges carry out various responsibilities in the district courts and often help prepare the district judges’ cases for trial. They also may preside over criminal misdemeanor trials and may preside over civil trials when both parties agree to have the case heard by a magistrate judge instead of a district judge. Unlike district judges, bankruptcy and magistrate judges do not exercise "the judicial power of the United States" but perform duties delegated to them by district judges. The judges on the U.S. Court of Federal Claims are also not Article III judges. Their court is a special trial court that hears mostly claims for money damages in excess of $10,000 against the United States. With the approval of the Senate, the President appoints U.S. Court of Federal Claims judges for fifteen-year terms. 

How many federal judges are there?Congress authorizes a set number of judge positions, or judgeships, for each court level. Since 1869, Congress has authorized 9 positions for the Supreme Court. It currently authorizes 179 court of appeals judgeships and 678 district court judgeships.(In 1950, there were only 65 court of appeals judgeships and 212 district court judgeships.) There are currently 352 bankruptcy judgeships and 551 full-time and part-time magistrate judgeships. It is rare that all judgeships are filled at any one time; judges die or retire, for example, causing vacancies until judges are appointed to replace them. In addition to judges occupying these judgeships, retired judges often continue to perform some judicial work.

And elsewhere on the site, this:

Law Clerks

The practice of hiring a recently graduated law student to serve as an in-chambers judicial assistant was pioneered by Horace Gray. Both as the chief justice of the Supreme Judicial Court of Massachusetts (1864–1881) and as an associate justice of the Supreme Court of the United States (1882–1902), Gray personally paid an assistant, whom he referred to as his “secretary.” Other justices of the Supreme Court followed the practice in the late-nineteenth and early-twentieth centuries. Although Congress in 1886 heeded the advice of the U.S. Attorney General that it pay for each of the justices to hire a stenographer “to assist in such clerical work as might be assigned to him,” it was not until 1919 that it provided funding for the hiring of legally trained assistants. To distinguish these assistants from the stenographers, Congress designated them as “law clerks.”

The early law clerks, most of whom were graduates of the Harvard Law School, conducted legal research, checked citations, and performed a wide range of personal and administrative tasks for their judges. Despite the concerns expressed by some members of the Judicial Conference that such assistance was unnecessary or that highly paid law school graduates were not needed to perform such tasks, Congress in 1930 provided funds for each circuit court of appeals judge to appoint a law clerk. Six years later, Congress authorized up to thirty-five district court judges to appoint law clerks, as long as the senior circuit judge of the circuit in which the district was located issued a certificate of need. A statute of 1945 lifted the restriction on the number of district court judges allowed to appoint a clerk. The certificate of need requirement continued until 1959, when Congress authorized judges to hire “necessary” law clerks subject to the limits of their chambers staff budgets and to the minimum law clerk salary provisions of the Judicial Salary Plan.

As federal judicial caseloads and budgets increased during the last four decades of the twentieth century, the number of law clerks retained by the judges of the federal courts rose steadily, though some judges have eschewed the practice of hiring short term law clerks in favor of “career” clerks, who are hired with the expectation that they will serve for a period of more than four years. Today’s law clerks typically perform quasi-judicial functions, such as preparing bench memoranda on legal issues and composing drafts of judicial opinions.

Further Reading:
Baier, Paul R. “The Law Clerks: Profile of an Institution,” Vanderbilt Law Review 26 (1973): 1125–77.
Newland, Chester A. “Personal Assistants to Supreme Court Justices: The Law Clerks,” Oregon Law Review 40 (1961): 299–317.
“Law Clerks: The Transformation of the Judiciary,” Long Term View: A Journal of Informed Opinion 3 (1995).