Open weights
A model whose trained parameters are published, so anyone can run it on their own hardware.
Also written: open-weight, open source model, self-hosted.
The weights are downloadable and you can run the model yourself. That is not the same as open source — the training data and code usually are not published, and the licence often carries conditions — so “open weights” is the more accurate term.
The reasons to want this are privacy, control, and not being subject to a vendor deprecating something you depend on. The cost is that you now operate inference (Running a trained model to get an answer. Everything you pay for by the token is inference, not training.) infrastructure, which is substantially more work than an API key and is rarely cheaper until volumes are high.