There are three parts to such web maps. A map service, A feature service (This bit is optional), and a client library.
The map service provides the map images and may be in a special service specific protocol, like in Google Maps, Bing Maps, or the main services provided by OpenStreetMap, but there are also standard protocols, particularly TMS and WMS which can be provided by various software like GeoServer. Some client libraries even let you just load one big image as is, though this doesn't work so well for very large maps or if you want a wide range of scales. TMS can also be handled by an ordinary web server is the map is static
The feature service provides the points, and other shapes overlaid on the map. Again there are special servers, or you can drop everything in a file and point the client at that file. GML, KML, and GeoJSON are common formats for feature data.
The client library is what ties everything together and makes the map interactive in the web browser. Google and Bing have their own special ones, and there are generic ones like OpenLayers and Leaflet.
The absolute simplest thing that would get you going would be to get OpenLayers or Leaflet, and try pointing it at a straight image of the map. You could improve on this by chopping the map into tiles to make a TMS, and by writing up your point features, etc in KML or GeoJSON files..