Hollister filed a "petition for certiorari" (a request that the Supreme Court hear an appeal) from the two lower courts (federal district court and federal court of appeals) which both bounced his eligibility lawsuit. The Government declined to respond to the petition (a response is not required in the Supreme Court; a failure to respond means the responding party--in this case, Obama-- doesn't think the Supreme Court has any interest in the case).
Once a Petition for Certiorari is filed, and either responded to or a response is waived, each such Petition is automatically "listed for conference." That does not necessarily mean that the Supreme Court will actually discuss the case; there are dozens-- sometimes hundreds-- of cases "listed" for each conference. Before the conference, each of the 9 justices circulates a list of cases they think are worth talking about. (These lists are private; the public never sees them.) If a case is on none of the 9 justices' lists, it is put on the "dead list" and is automatically denied without the court actually "discussing" it. (The list of denials is published the next business day after the conference. No reason is stated for a denial.).
If any of the 9 justices thinks a case is worth talking about, but the responding party didn't file any response to the Petition for Certiorari, the Court will usually ask the other party to respond, and put the case over for another conference.
If four of the nine justices think a case should be heard, the Court will grant the petition and schedule the case for full briefing and argument. If less that four want to hear the case, the petition will be denied without stating any reason, but the few who wanted to hear it can (but don't always) file a dissent from the denial. (If the case was on the dead list, then no one will dissent).
My prediction: the court will not ask for a response, and the petition will be denied without dissent.
Excellent summary. Thank you.