According to September 2007 PC World, the ability to have web applications which work offline is coming, but not quite here yet. PC World points out that many of the applications don't allow the user to seamlessly switch from offline to online and have more limited features than pure online services (which are themselves much more limited than the typical PC application).
Firefox 3 is planned to have an offlineResources attribute to allow the web designer to tell the browser what should be cached for offline use. The Google Reader feed reader now has an offline options, which requires Google Gears to be installed on the user's PC. Dojo Offline is an toolkit to make Google Gears easier to implement.
The same issue of PC World looked at web based office packages, such as Google Docs and Spreadsheets, ThinkFree and Zoho. Currently these are purely online: you have to have an Internet connection to work on documents, or export the work to a PC based package such as OpenOffice.org or MS-Office.
Also in PC World were more specialised business applications available online, such as HighRiseHQ and FreeCRM Customer Relationship Management systems (CRMs). These are also online only systems at present. Also of interest is the AgoraCart Open Source e-commerce shopping cart software, although this is downloaded software, not an online system hosted on a host web site.