Producer's Risk is the risk of making a Type I Error - rejecting the Null Hypothesis when it is, in fact, true.
If you think of the final inspection of a product in a factory, why this is called Producer's risk becomes clearer. The null hypothesis for the inspection process is that the product is acceptable. If the product is found to be defective when it is actually not defective, it will be sent to repair, or discarded when it shouldn't have been. This causes the factory - the producer - to incur unnecessary costs because they will rework or scrap a product that is actually acceptable.
When you set the alpha level for a hypothesis test, you are setting the level of producer's risk you are willing to accept.