What are the requirements or preconditions that must be met in order for someone to file an Amicus Curiae "friend of the court" brief?
What form can the brief take?
Some United States Supreme Court rules are helpful, though these rules are not stated as applying to any other court.
The brief must be submitted within a certain time period, depending on the case at hand:
An amicus curiae brief in support of a petitioner or appellant shall be filed within 30 days after the case is placed on the docket or a response is called for by the Court, whichever is later, and that time will not be extended. An amicus curiae brief in support of a motion of a plaintiff for leave to file a bill of complaint in an original action shall be filed within 60 days after the case is placed on the docket, and that time will not be extended. An amicus curiae brief in support of a respondent, an appellee, or a defendant shall be submitted within the time allowed for filing a brief in opposition or a motion to dismiss or affirm. An amicus curiae shall ensure that the counsel of record for all parties receive notice of its intention to file an amicus curiae brief at least 10 days prior to the due date for the amicus curiae brief, unless the amicus curiae brief is filed earlier than 10 days before the due date.
The brief must be submitted electronically and in print:
An electronic version of every amicus curiae brief in a case before the Court for oral argument shall be transmitted to the Clerk of Court and to counsel for the parties at the time the brief is filed in accordance with guidelines established by the Clerk. The electronic transmission requirement is in addition to the requirement that booklet-format briefs be timely filed.
Wikipedia claims that the brief must be submitted as a printed book in forty copies, but I can not substantiate that within the Supreme Court rules - which is the document cited! Other briefs are required to be submitted in print in forty copies, so such a requirement would not be overly surprising, but I don't know if this is true.
This states more requirements for general courts:
1) if the amicus curiae is a corporation, a disclosure statement like that required of parties by Rule 26.1;
(2) a table of contents, with page references;
(3) a table of authorities—cases (alphabetically arranged), statutes and other authorities—with references to the pages of the brief where they are cited;
(4) a concise statement of the identity of the amicus curiae, its interest in the case, and the source of its authority to file;
(5) unless the amicus curiae is one listed in the first sentence of Rule 29(a), a statement that indicates whether:
(A) a party's counsel authored the brief in whole or in part;
(B) a party or party's counsel contributed money that was intended to fund preparing or submitting the brief; and
(C) a person—other than the amicus curiae, its members, or its counsel—contributed money that was intended to fund preparing or submitting the brief and, if so, identifies each such person;
(6) an argument, which may be preceded by a summary and which need not include a statement of the applicable standard of review; and
(7) a certificate of compliance, if required by Rule 32(a)(7).
There is a length limit:
Except by the court's permission, an amicus brief may be no more than one-half the maximum length authorized by these rules for a party's principal brief. If the court grants a party permission to file a longer brief, that extension does not affect the length of an amicus brief.
There is also a slightly different timeframe:
An amicus curiae must file its brief, accompanied by a motion for filing when necessary, no later than 7 days after the principal brief of the party being supported is filed. An amicus curiae that does not support either party must file its brief no later than 7 days after the appellant's or petitioner's principal brief is filed. A court may grant leave for later filing, specifying the time within which an opposing party may answer.
In addition to all of this, a guide to filing amicus curiae briefs can be found here. It should, of course, only be used as a basic introduction to the briefs.