01In plain English
An open-weight model is one whose trained parameters, its "weights", are published for anyone to download and run. That is not always the same as open source: the training data and code may still be private, and the licence may limit how you use it.
02Why it matters when you are choosing
Open weights let a company run a model on its own servers, keep data in-house and avoid depending on one provider.
03What to check
Ask the vendor
Read the licence for commercial limits, and check what hardware you need to run the model well.