Anybody who played the original Destiny will be familiar with public events. These are challenges which appear on your map at specific time intervals. In Destiny 2 they are much easier to find as they are marked on your map a few minutes in advance, along with the amount of time until they begin.
Simply approch the flag, and hold x to join the public event. You will be given certain tasks, such as defending a point, killing specific enemies etc. In Destiny 2, you can trigger a much harder "heroic" version of these events by performing certain actions.
Ill try to update this as much as possible as and when heroic events are discovered, but there is a simple one at the very beginning of the game. Directly outside the church where Devrim is located in the EDZ, a public event spawns to kill fallen extractors who are mining glimmer. A glimmer drill appears with waves of enemies. Instead of killing the enemies, destroy the small drilling machines around the area. They glow bright blue, are knee hieght and should be fairly obvious. Doing this will cause the message "heroic event started" to appear, and change your objective to defending the glimmer pile. You will have to fight several waves of fallen and cabal while you wait for the glimmer to transmat, including a mini boss. If you die, you will respawn in the church a few seconds later to rejoin the action.
There is a public event on titan in which you have to destroy a fallen walker. To access the heroic version of this event, simply destroy a leg. This will cause the walker to drop three arc orbs, which must be placed into containers around the event which hold scorch canons. There are a total of 6 orbs required, meaning you will have to down the walker twice and place all 6 orbs before destroying it. Activating the heroic event spawns a second walker. You must defeat both within the time limit given.
There is an event in which you have to capture a cabal drill. During this event, a thresher ship will fly around the area. It usually appears when you are 40% complete, then again at 80%. Simply destroy this ship to spawn a large boss and activate the heroic version.
On Nessus there is a public event to defend a conflux from vex sacrifice. Around the area will be a couple of sync plates. They are round circles on the floor which fill with light as you stand on them, and should look familiar to anybody who played the vault of glass raid in Destiny 1. You have to stand inside these plates to capture them, and any vex walking onto the plate will stop your capture. After you have taken both, the event will become heroic. The goal remains the same - stop vex sacrifices - but the enemies become more powerful.
Upon completion, a reward chest will appear, and your achievement should pop a few seconds after collecting your loot. In destiny 2 any participation in a public event grants some reward, but it is important to note that for this achievement you MUST fully beat the event.