With an open-weight model, the developer publishes the trained weights so that anyone who accepts the license can download the model and run it on their own hardware. This lets organizations host it privately, adapt it with their own data and inspect how it behaves, without sending data to a provider.
Open weights are not the same as open source. The weights may come with a license that restricts commercial use, limits use above a certain scale or forbids certain applications. The training data, training code and the methods used are often not released, so it may not be possible to reproduce the model.
There are trade-offs. The user takes on the work of hosting, updating and securing the model. Once it is released, the developer cannot recall it, and safety measures built into the model can sometimes be removed by those who fine-tune it.
Read the license carefully, check what was and was not released, and test the model on your own tasks before relying on it.