What GIS software is used by web maps sites (Google, Bing Maps, etc)?

Google use a self-developed tool called Atlas to maintain geodata.

In this video from Google I/O 2013 you see how Atlas works (Atlas starts at 7:30 - but it is interesting to see the whole video).

You didn't specifically ask about any open source tools or data in your original post, but OpenStreetMap has a global web map that is as good as the ones from Google or Bing. Because it is open source, there are lots of great resources explaining how they manage their data and render their map.

If you want to create a worldwide map like Google or Bing, you can use OpenStreetMap data and their open source tools to do it. You'd store your data in PostgreSQL/PostGIS, render map tiles using Mapnik, and serve those map tiles on the web using something like Apache and mod_tile. Any web page that wanted to include your map could do it with OpenLayers or Leaflet (which take the place of something like Google Maps API).

Switch2OSM has some good instructions for how you would go about setting up your own global map.