A Galaxy Event is a unique scenario, typically lasting for about 5 days, in which Starfleet must assist in a crisis requiring the creation and delivery of supplies. Multiple Factions will have a stake in the result, and though initial efforts may be mutually productive, you may want to decide who you might support in case of a breakdown in cooperation.
During a Galaxy Event, you will be running Supply Missions, an activity that is unique to this type of scenario and can be accessed via the Event menu itself. The objective of these missions is to build the supplies needed to avert the impending crisis. As such, you may need to gather the components to build those supplies via traditional means if you don't already have them. Once you're ready to build the supplies in question, you'll need to assign a proficient Crew member to complete the task. Doing so will grant you Victory Points, which are used to determine your Rank in the Event, as well as what personal Threshold Rewards you will earn along the way. Each time you complete a Supply Mission, you also have a chance to obtain a Super-Rare Item that can be contributed directly to the relief efforts for additional Victory Points.
Galaxy Events are divided into two Phases. Phase One is typically for the good of the many, and during this time you can collectively work toward unlocking Community Rewards that everyone can benefit from. But Phase Two tends to be when relations break down between the involved factions, and as a representative of Starfleet you will ultimately have to take a side. At the end of the event, the winning faction will provide a small bonus to those who supported it.
To participate in an Event, tap the temporary tab on the left side of the screen to view the details, and then tap "Enter Event". Events become accessible at level 8.