This is a fascinating, and by no means simple process.

In England a coat of arms can only be granted by “The College of Arms” after a fairly detailed process

First a person or institution needs to petition for arms. If the Earl Marshal approves a petition he will issue his Warrant to the Kings of Arms allowing them to proceed with the grant. At this stage the designing of the arms will begin. The Kings of Arms have full discretion over the design of the armorial bearings they grant, but the wishes of the applicant are taken into account as fully as possible. Once the design has been agreed with the petitioner it is checked against all previous arms on record to ensure it is distinct and then submitted to the Kings of Arms for their approval it is signed and sealed by the Kings of Arms, and a copy of it painted and scrivened into the official College registers.

Everything you need to know about the process and some fascinating information about the College of Arms can be found on their website