Here's a quick shot at the Cliff's Notes version (quotations from the brief italicized):
Whether the Second Amendment guarantees lawabiding, adult individuals a right to keep ordinary, functional firearms, including handguns, in their homes.
Mr. Heller officially responds to DC's petition (re-joined by the other 5 original Parker petitioners on the grounds that they were a party to the case in lower courts), stating "The Court should grant certiorari on the question as presented by Respondents." ('Certiorari' = 'take the case')
Summary of Argument: "the people" means "individuals in our country". High courts are divided on the issue, and SCOTUS must resolve the split. The case, properly presented, will be very simple: does the 2ndA protect an individual right, and do the two DC laws in question absolutely negate that right? (Expected answer: yes.) BTW: DC is way outta line in how they presented the issue, so the question they presented should be replaced with Heller's (above).
I. The Court Should Provide Essential Guidance to Courts That Misconstrue the Second Amendment: it's an individual right, not collective. The people are all individuals, not a select government-chosen body. The militia is members of the public capable of bearing arms in defense of the government if called upon to do so. Miller was very simple and clear on this - but somehow courts have twisted that into the opposite. The whole issue has gotten scrambled, and SCOTUS should explain how the 2nd Amendment should be interpreted.
II. The Question Presented by Petitioners Misrepresents the Court of Appeals’ Holding and the Central Issue in This Case: DC says X may be banned because Y is allowed - but fails to explain that, for all practical purposes, Y is not allowed either. SCOTUS needs to clarify what the question actually is, and what the answer to it means. (While well done, the wording of this section makes me nervous, as SCOTUS could say "fine...DC, try again" or worse "crappy question, next case" - ctd.)
III. Whether the Second Amendment Forbids Handgun Prohibition Is a Discrete Constitutional Question That Must Be Answered on Its Own Terms: Miller addressed the question of whether Z was "militia suitable"; handguns are plainly "militia suitable", and DC's notion that they are not is flatly unsupportable, social commentary notwithstanding. The two bans together form such a comprehensive total prohibition that bickering over the division between the bans is pointless until both bans have been firmly overturned.
IV. The City’s Policy and Social Science Arguments Are Irrelevant and Factually Baseless: DC claimed the ban is effective, when facts easily and plainly show it is not - and that is, nonetheless, irrelevant because this is a court of LAW, not of policy or sociology.
V. Citizens Under Criminal Attack Are Not Required to Stand By and Die Awaiting Police Protection: In claiming the city is not required " to stand by while its citizens die", the city perversely requires its citizens to do so despite numerous court cases where the city fought for the right " to stand by while its citizens die" (to wit: DC is not obligated to protect citizens, both by the laws of this land and the laws of physics).
Overall the brief is good, systematically demolishing everything DC presented to justify its side.
My one concern is whether it is too good: SCOTUS could say "this is a mess, we'll let the lower court ruling stand until someone else presents a better challenge."
I think SCOTUS will take the case, precisely because there is a legal mess that only they can clear up, SCOTUS justices rarely get a case this profound & fundamental & wideranging (i.e.: they can't pass up a case this good), and it really can be resolved with a simple, narrow, yet profoundly impacting verdict.