Essentially, there are two types of editing. You can edit features you've added to the map, like shapefiles and CSVs or you can edit Feature Services hosted in ArcGIS Server. For shapefiles and CSVs, you need to enable editing by clicking the arrow next to the name of the layer when viewing the Map Contents panel. You will then see an Edit button on the menu bar, allowing you to edit the features. Note that only you can edit the features and save it back to the map. While other people who view your map could similarly enable editing, they won't be able to save the changes to your map because they don't own it. They can only save their own copy. In the case of editing features in a Feature Service, everyone will be able to edit the service.
