How do graduate students apply for patents? What rights and obligations do my tutor and I have?

Your situation should be a service invention, and its application right (that is, the applicant) should belong to your university. You and your tutor are inventors of the invention, not applicants. Can only apply in the name of the school. If you make an individual application without the consent of the school, you may be sued by the school.

See the definition of service invention and non-service invention in Article 6 of the Patent Law for details:

Article 6 Inventions and creations made by performing the tasks of the unit or mainly using the material and technical conditions of the unit are service inventions and creations. The right to apply for a patent for a service invention-creation belongs to the unit; After the application is approved, the entity shall be the patentee.

The right to apply for a patent for a non-service invention-creation belongs to the inventor or designer; After the application is approved, the inventor or designer shall be the patentee.

Where an invention-creation completed by making use of the material and technical conditions of the entity has a contract with the inventor or designer to stipulate the right to apply for a patent and the ownership of the patent, such agreement shall prevail.