While still at Apple, Steve Jobs was shown the Smalltalk work that was going on at Xerox PARC, and a lot of ideas in the NeXT system came from there. Although Smalltalk was the first environment to adopt the idea of object-oriented development, NeXT pushed the concept commercially.
The NeXT Application Kit encapsulated every user interface component in an object, and was largely responsible for a number of software patterns that are now so ingrained in developers' minds that it seems strange to think that someone even needed to invent them.
Using objects to represent user-interface components is now completely standard. Whether you're on UNIX, Windows, or any other modern GUI, user interface elements are objects. Even in languages that don't directly support object orientation, such as C or Haskell, some (pseudo) object-oriented style is commonly used.
Perhaps the most-copied idea from NeXT was the Interface Builder (IB). Sadly, this is an example where everyone copied the superficial concepts, but missed the main point.
The obvious improvement of NeXT's Interface Builder over other systems was that the IB let you draw user interfaces. A graphics designer could use IB to draw what he thought the application should look like. This drawing then became the interface; there was no need to re-create it in code or markup. Systems like Microsoft Visual Basic and Borland Delphi copied this part of the system very well.
NeXT went one step beyond this design, however. IB didn't just let you draw interfaces; it let you create objects and define relations between them. Some of these objects were the views that were displayed onscreen. Others might be controllers, or even the occasional model object. The files that the Interface Builder generated were serialized forms of these object graphs. You could instantiate a copy of one with a single line of code, and you would get back a window, all of the views inside it, all of the controllers, and typically just a single connection point to connect them to your model.