Classic mode (the only mode in IIS6 and below) is a mode where IIS only works with ISAPI extensions and ISAPI filters directly. In fact, in this mode, ASP.NET is just an ISAPI extension (aspnet_isapi.dll) and an ISAPI filter (aspnet_filter.dll). IIS just treats ASP.NET as an external plugin implemented in ISAPI and works with it like a black box (and only when it's needs to give out the request to ASP.NET). In this mode, ASP.NET is not much different from PHP or other technologies for IIS.
Integrated mode, on the other hand, is a new mode in IIS7 where IIS pipeline is tightly integrated (i.e. is just the same) as ASP.NET request pipeline. ASP.NET can see every request it wants to and manipulate things along the way. ASP.NET is no longer treated as an external plugin. It's completely blended and integrated in IIS. In this mode, ASP.NET HttpModules basically have nearly as much power as an ISAPI filter would have had and ASP.NET HttpHandlers can have nearly equivalent capability as an ISAPI extension could have. In this mode, ASP.NET is basically a part of IIS.