The Red Hat is a hat in Nintendogs. The M stands for Mario, a classic character in Nintendo games and also Nintendo's mascot. Its counterpart is the Green Hat, which is Luigi's. It falls under the Accessories category, and can be worn by dogs. It is not available in Nintendogs + Cats. It can be obtained by walking a dog and finding it as a present. It sells for $20 on Nintendogs: Best Friends.
DS: "This hat is standard gear for mustachioed master plumbers."