The weights are what training produces. They are a very large set of numbers, stored in files, that together make up the working model. To use a model, software loads the weights into memory and runs inputs through the network with them. The architecture describes the shape of the network, and the weights hold what it learned.
Because they hold the result of very expensive training, they are valuable. Companies guard them closely for closed models, using access controls and monitoring, since stolen weights could be copied, run without limits and stripped of safety measures. For open-weight models, the files are released publicly under a license.
The files can be large, from a few gigabytes for small models to hundreds of gigabytes or more for large ones, and they come in several formats and precisions. Smaller versions made by quantization trade some quality for lower memory use.
If you download weights, check the source and the license, verify the files, and keep them secured if they are proprietary.