If you just need a simple overlay with some markers or routes then you can use Leaflet or OpenLayers.
It is also possible to create a completely personal map, but it will require to set up a complete backend. iD is just the editor. If you want to see the resulting map in you browser you will need:
- a postgres database for storing the map data
- a renderer for creating images (tiles) from this database, for example Mapnik
- a java script library like Leaflet or OpenLayers for creating a browsable map from the rendered tiles
- possibly a geocoding service like Nominatim if you need a search function
As you can see, OSM's main goal is not to create personal maps. Still it is possible but requires some effort to set it up (switch2osm has some useful guides).
Also take a look at similar questions on help.openstreeetmap.org, including possible alternatives:
- https://help.openstreetmap.org/questions/27621/personalised-maps
- https://help.openstreetmap.org/questions/22827/using-id-editor-to-for-personal-alternative-maps
- https://help.openstreetmap.org/questions/11069/does-openstreetmap-have-a-my-places-feature-like-google-maps
- https://help.openstreetmap.org/questions/24591/create-my-own-addresses-roads-and-ways-on-a-custom-world-map
- https://help.openstreetmap.org/questions/4496/how-to-create-fantasyimaginary-maps-with-openstreetmap