Except in rare circumstances, cases are brought and shepherded through the Court by members of the Supreme Court bar; appeals on behalf of the United States are handled by the solicitor general.
The Supreme Court lays out rules for briefs and oral arguments:
Petition Stage (See Rules 12, 14, 15, 33, 34, 37):
Once a petition is received, cases are put on the docket. (Most cases have this docket # format: term year-number. E.g., 06-984. The small number of cases that fall under the Court's original jurisdiction, rather than appellate, have a number followed by "Original" or "Orig.", or an upper or lower case O--not zero--separating two parts. E.g., No. 143 Original or 22o143. For more on docket numbering, see the National Archives and the Supreme Court Database.) The word docket generally refers to all petitions seeking review, but can also mean all those petitions accepted for review. All petitions accepted for review are also known as the plenary docket or the decision agenda. Some proceedings may be in forma pauperis.
SCOTUSblog has a list of cases ("Petitions We're Watching") it believes have the best chance of being granted in advance of each conference. If the Court grants a petition, the appellant has 35 days to file briefs and the Court generally schedules an oral argument within 3 months. (Cases granted review are not necessarily heard that same year).
Merits Stage (See Rules 24, 25, 26, 33, 34, 37):
Oral Arguments held (See Rule 28). (Justices also have the option to “summarily reverse” – i.e., overturn a lower court ruling without hearing oral arguments.)
Why is a case waiting for cert relisted?
A Conference may include cases that were “relisted” – i.e., considered in Conference the previous week, then held over to be discussed again. A case can be relisted for many reasons, including because a Justice makes that request, because the Justices are preparing an opinion summarily reversing, and because one or more Justices is preparing an opinion dissenting from the denial of certiorari. A majority of the relisted cases are ones in which a state has petitioned arguing that the court of appeals should not have granted relief to a prisoner on “habeas corpus,” which is a way of bringing a challenge to a conviction in federal court.
What is the difference between "held" and "relisted?" "Relisted" means that they will consider it at the next Conference, at least in theory. "Held" means that it is not considered at a conference until it shows up as being distributed again; usually this means that the Court is waiting to reconsider it after it issues an opinion on a similar question, although a cert. petition can also be held while the Court waits for a petition presenting the same question to catch up.
When will a big decision come down?
When the Court is hearing arguments, it usually releases orders on Monday and opinions on Tuesday (and possibly Wednesday). Once scheduled oral arguments are over sometime in the spring, the Court only sits to release opinions, and it will usually do so on Monday.
Wondering when to file a petition for cert?
There are certain times of year when the Court is making more of an effort to fill its calendar, especially in early January (to make sure it has enough cases for the rest of the Term) and in late June. The late June spike occurs because it wants to make sure that it has enough cases to fill up the fall. When the Justices are on their summer recess, they usually don't grant new cases until late September. (Although there isn't generally enough time to grant a case in late September and have it briefed in time for argument in early December.)
There is no specific deadline for the Solicitor General to file briefs when invited to do so by the Court, but the Solicitor General tends to file invitation briefs (in cases in which the Court has asked the federal government to weigh in) in batches in late November/early December and then again in late spring. Thus the number of grants by the Court is also high in January and late June because the Solicitor General tends to file his invitation briefs in time for those cases to be considered during early January and late June.
The idea of the Court granting more cases in early January and late June is now sufficiently well-known, and enough lawyers are trying to get their cases on the Court's Conferences around that time, that you may not be improving your odds by that much anymore by moving a case to a more "desirable" Conference. The lawyers who are opposing cert. can also make efforts to try to ensure that the cases are NOT on those conferences, either by taking an extension or filing a response early.