How does the FAA define day and night?
Day and Night for Pilots

As pilots we often talk about night flying and daytime flying meaning when it is dark or light, but for logging time and for currency there are specific definitions that we must pay attention to.

FAR 1.1 General Definitions

Night means the time between the end of evening civil twilight and the beginning of morning civil twilight, as published by the American Air Almanac, converted to local time.

Day is not defined in the FARs but the definition of night implies that it is the time between the beginning of morning civil twilight and the end of evening civil twilight.

The Air Almanac is published annually on CD-ROM but you can find sunset, sunrise, and civil twilight times for specific locations at the US Naval Observatory website.

The time between sunrise/sunset and twilight is 26-30 minutes depending on the time of year. Since you can easily find sunrise/sunset information in newspapers, the weather channel, etc. you can estimate twilight fairly easily.

Currency
§61.57 (b) (1) … no person may act as pilot in command of an aircraft carrying passengers during the period beginning 1 hour after sunset and ending 1 hour before sunrise, unless within the preceding 90 days that person has made at least three takeoffs and three landings to a full stop during the period beginning 1 hour after sunset and ending 1 hour before sunrise, and…

VFR Weather Minimums
Only Class G airspace has different minimums depending on night and day and they use the standard night definition—end of evening twilight to the beginning of morning twilight.

Aircraft Lights
Aircraft lighting is required between sunset and sunrise. 
§91.209 During the period between sunset and sunrise no person may 1. operate an aircraft unless it has lighted position lights, 2. park or move an aircraft in an area of an airport where night operations are being conducted unless it is clearly illuminated, has position lights, or is in an area marked by obstruction lights.

Aeronautical Experience Requirement
In order to obtain a private pilot certificate night training is required. The commercial pilot certificate requires solo night flights. In these cases the standard night definition—end of evening twilight to the beginning of morning twilight—applies.

Special VFR
AIM 4-4-6 g. Special VFR operations are prohibited between sunset and sunrise unless the pilot is instrument rated and the aircraft equipped for IFR flight.

Logging Night Flying Time
Logging of night flight should be made in accordance with Part 1 definitions, which is from the end of evening civil twilight to the start of morning civil twilight. This differs from the definition used for night currency.

Summary

Position lights – sunset to sunrise

Special VFR – sunrise to sunset unless instrument rated

Carry Passengers – 1 hour after sunset to 1 hour before

Everything else – twilight to twilight