open-weight model
A model whose learned numbers are published, so anyone can download and run it themselves.
A model's weights are the numbers it learned in training. Some companies keep them private and only offer the model through their API. Others publish them as an open-weight model. You can then run it on your own computers, keep your data in-house, and adapt it. You also take on the work of running it.
Think of it as
Like being given the recipe and ingredients to cook at home, instead of only being able to order the dish from the restaurant.
Example
A hospital cannot send patient notes to an outside service. It downloads an open-weight model and runs it on its own servers, so the notes never leave the building.