Stable Diffusion's VAE is a neural network that encodes and decodes images into a compressed "latent" format. The encoder performs 48x lossy compression, and the decoder generates new detail to fill in the gaps.
(Calling this model a "VAE" is sort of a misnomer - it's an encoder with some very slight KL regularization, and a conditional GAN decoder)
This document is a big pile of various links with more info.
- CompVis