The nursery in "The Veldt" is a very large room that displays interactive, immersive computer simulations that respond to and reflect the thoughts of the participants. In short, it is like an advanced form of virtual reality.
David McClean, the psychologist in the story, seems to imply that the nursery technology originally was used as a therapeutic tool rather than a consumer product; the intention was to observe the "patterns" that the child interacting with the computer produced in order to gain an insight into trauma. However, since the technology has apparently become commercialized, it has instead taken a turn for more entertainment-based functions, and has essentially replaced nearly all other forms of leisure for the Hadley children, including their relationship with their parents.
The nursery is operated by a computer, whose technology is implied to be more advanced than our own. The computer is capable of obeying both thought commands and spoken ones, and can create a wide variety of environments, which it displays on the walls of the nursery. The illusion is supported by chemical scents and weather, as indicated by the sun causes characters to sweat.
The ending of the story does not clarify whether the Hadley's nursery was malfunctioning, or if Peter and Wendy had somehow hacked it to respond only to their commands, or to alter its functions. However, the nursery does not respond to their father's orders, and it is somehow able to make the lions tangible enough to kill their parents, which seems to defy the idea that the images are all "behind glass screens" as Mr. Hadley had stated.