Both Geoserver and MapServer are geospatial servers that can export data in one of the many formats consumable by mashups like OpenLayers. They can be backed by data in shapefiles or spatial databases like PostGIS. They also both support the WFS-T protocol which allows you to update the data on the server from a desktop client. As far as I am aware, ESRI software doesn't support WFS-T, but there are several open source and commercial products that do. Here is a a list: https://gis.stackexchange.com/questions/347/how-many-wfs-t-clients-exist
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