Considerable observational efforts are being dedicated to measuring the sky-averaged (global) 21-cm signal of neutral hydrogen from Cosmic Dawn and the Epoch of Reionization. Deriving observational constraints on the astrophysics of this era requires modeling tools that can quickly and accurately generate theoretical signals across the wide astrophysical parameter space. For this purpose artificial neural networks were used to create the only two existing global signal emulators, 21cmGEM and globalemu. In this paper we introduce 21cmVAE, a neural network-based global signal emulator, trained on the same dataset of ~30,000 global signals as the other two emulators, but with a more direct prediction algorithm that prioritizes accuracy and simplicity. Using neural networks, we compute derivatives of the signals with respect to the astrophysical parameters and establish the most important astrophysical processes that drive the global 21-cm signal at different epochs. 21cmVAE has a relative rms error of only 0.34 - equivalently 0.54 mK - on average, which is a significant improvement compared to the existing emulators, and a run time of 0.04 seconds per parameter set. The emulator, the code, and the processed datasets are publicly available at https://github.com/christianhbye/21cmVAE and through https://zenodo.org/record/5904939.