The Night button changes SkySafari's appearance to a red-on-black theme designed to help preserve your night vision. Night Vision is best used in the dark, out under the stars. Tap this button once to switch to the Night Vision theme; tap again to restore SkySafari to its previous appearance.
For many people, the screen is still too bright to effectively preserve their night vision - even when SkySafari is using its Night Vision theme.
The iOS version of SkySafari has a Screen Brightness setting (also found in the Settings > Appearance & Behavior view) which lets you adjust the screen brightness. However, this setting only works within SkySafari. To turn down the screen brightness for all apps on your devices, use the screen brightness slider in the main Settings app on your device.
An even better solution is to place a piece of red film over the screen. A brand called Rubylith works particularly well. A "hardware" approach to preserving your night vision works better than any software solution because it will enforce a red appearance across all applications, not just SkySafari.