Notice of motion vs. order to show cause
A notice of motion is the ordinary vehicle: you pick a return date consistent with the required notice period, serve the papers, and the other side responds. An order to show cause is used when you need relief quickly or need to alter the normal briefing schedule — a temporary restraining order, a stay of enforcement, or a request to vacate a default. It requires judicial signature before service.
What a motion record looks like
An attorney affirmation, an affidavit from someone with personal knowledge, the exhibits that prove your point, and a memorandum of law applying the standard to those facts. Attorney statements about facts carry no evidentiary weight — the affidavit does the work. Exhibits must be authenticated. A motion supported only by argument tends to lose.
Summary judgment
The movant must make a prima facie showing that there is no material issue of fact requiring a trial. Only then does the burden shift to the opponent to raise one. Because the standard is demanding and the record must be complete, summary judgment usually comes after discovery closes, and deadlines to file it after the note of issue are strictly enforced.
Read the judge's rules
Brooklyn judges publish part rules covering page limits, courtesy copies, adjournment practice, pre-motion conference requirements and appearance procedures. Ignoring them is the fastest way to have a well-argued motion marked off or rejected. Check the part rules before every filing — they change.