The 'weights' are the trained internal numbers that make a model work. Open weights means a company publishes those numbers so anyone can download the model, run it themselves, and build on it. Closed means the model stays locked on the company's servers and you can only reach it through their service. Open offers freedom and control, while closed often offers convenience and tighter oversight.
For example, You can download an open weights model and run it on your own computer with no internet connection.