A web service takes something from your GIS and puts it in the hands of people who may not have direct access to your organization's data. When you expose your GIS as web services, you're making it available to a variety of apps that might be running on desktop workstations, tablet PCs, or mobile smartphones. You can choose to expose your web services only within your office, but you also have the flexibility to make your web services available to any device that can connect to the Internet.
ArcGIS offers various ways to make your GIS available through web services. These options include ArcGIS for Server, the Spatial Data Server, and ArcGIS Online. The web services you publish through these technologies use standard architectures such as REST. In fact, Esri has openly published the communication patterns used by its REST-ful web services in a document called the GeoServices REST Specification.
Esri freely distributes application programming interfaces (APIs) for popular frameworks such as Adobe Flex and Microsoft Silverlight that help you build apps with your GIS web services. You can easily blend a variety of web services from your GIS and from other sources to make an intelligent web map. For example, you might publish your own bus route data through an ArcGIS for Server web service and overlay it with a detailed Bing Maps basemap in a web application.
Traditional GIS applications such as ArcMap can also use your web services. However, because the services work through standard web service protocols and architectures, you aren't required to use ArcGIS technologies to consume them.