The World Wide Web
Sir Tim Berners-Lee wrote the first web browser, called WorldWideWeb, on his NeXTcube. He is quoted as saying that he would not have been able to do it if the NeXT development environment had not made it so easy. So a NeXT machine was both the first web browser and the first web server.
If you look at any modern graphical web browser, you'll still see a lot of user interface elements in common with WorldWideWeb. The relationship goes even deeper than that, however. The original HTML 1.0 tags all corresponded directly to things that the NeXT Application Kit's text system could handle easily. If you wrote any HTML before HTML 4.0 rearranged everything, odds are you were still using tags designed to be easy to render on a NeXT system.
NeXT was a very early adopter of web technology. For example, WebObjects wasn't quite the first web application framework; it was actually the second — by about two months. It wasn't massively popular, although companies such as Disney and the BBC used it, but it contained a lot of concepts that other frameworks later adopted.
WebObjects was popular with companies that had large databases that they wanted to expose to the Web, thanks to WebObject's integration with NeXT's Enterprise Object Framework (EOF). This integration provided a mapping between objects and data in a relational database, allowing you to expose this data to the Web easily.
Two open source projects have since re-created WebObjects 4.5, the final version to be written in Objective-C: SKYRiX Object Publishing Environment (SOPE) and GNUstepWeb. (Subsequent versions were in Java.) Both are still used commercially. SOPE is the foundation of projects like OpenGroupware.org, and is capable of scaling to a very large number of clients.
The Computer Is Not the Data
NeXT computers were very expensive. They were designed for multiuser environments, such as research labs where the computer would be shared among several students or researchers. The original units used a magneto-optical drive. This was a 250MB disk drive — large enough to store a copy of the operating system and all of the user's files. A typical configuration would have one disk per user. When you wanted to use a NeXT system, you would find a spare one, insert your disk, and boot from it.
This system might be very familiar to users of live CDs, or people who carry around a virtual-machine image on a USB flash drive. Live CDs have to work around the limitations of the hardware (typically by storing user files on a separate flash drive or in RAM, or copying them to a remote server), but the NeXT version allowed you to use a single, rewritable disk. You can do something similar on newer computers that can boot from USB or FireWire, but it's not very common.