From Destinypedia, the Destiny wiki
A Public Event is a type of encounter in Destiny in which several players in the same area can work together to achieve a common goal. Events may occur while on Patrol or within Strike missions.
Public Events are rated after completion based on the Guardians' performance; if time runs out before the objective is met, it is a Bronze-Tier rating. If time runs out but the Guardians manage to almost complete the objective, it is a Silver-Tier rating. If the objective is completed before the timer runs out, is a Gold-Tier rating. Guardians are rewarded with Glimmer, Motes of Light and experience instantly. Earlier in the game's history, it would reward Ascendant Shards or Ascendant Energy for Gold-Tier ratings. Guardians could pick up their rewards at the Postmaster if it was their first Gold-Tier rating of the day.
Destiny Public Event types
Destiny Public Event schedule
Note: Sometimes instead of a Public Event, the player will receive the message "The enemy is moving against each other", and a large group of two types of enemies will start fighting. Most of the enemies found in these events will be very powerful (i.e. Hallowed Knight, Reaver Captain). Fighting in this event does not reward the player with any bonus experience. This will happen in the Skywatch, The Barrens, and Ishtar Commons. Also, the player might receive the message "A roving band of Pikes is nearby." In this instance, several Dregs on Pikes will appear. This can happen at the Forgotten Shore as well as on the Moon.
Additional Note: When fighting Hive Wizards for the Kill an Enemy events the Wizard will sometimes "teleport" near the end of the event if the players manage to slow them down sufficiently for an extended period, usually resulting in failure of the event due to the time taken to re-engage the target.
Destiny 2 Public Event types
In Destiny 2, Public Events appear more frequently than in Destiny and are also announced 5 minutes before they begin. Once an event is announced, an icon will appear on the map screen for the planet, which allows players to find and track the event. If a player arrives at the location before the start of the event they will find a glowing flag and have the option of rallying to it. Rallying to the Flag instantly recharges all of their abilities, including their Super, refills all ammunition, and displays the time until the event starts. Successfully completing the event will reward participating players with Glimmer, a piece of equipment, and planetary materials that can be turned into that planet's vendor to increase reputation. Failing an event still yields materials and Glimmer but does not give equipment. Additionally, Public Events now have a Heroic version that is automatically unlocked once certain objectives are met during the event. The Heroic version increases the difficulty of the event but also increases the value of the rewards upon completion, such as a chance of receiving multiple pieces of equipment and/or a Legendary or Exotic Engram.
While the above mentioned Public Events start automatically in regular intervals, Protocols have to be started by players.
List of appearances