When a student was Muggle-born the acceptance letter was delivered in person by a member of the Hogwarts faculty. The faculty member acted as a special messenger, who explained everything to the student and parents.
The letter is written on parchment paper and delivered to the recipient by owl, except in the case of Muggle-borns and magical children living with Muggles (such as Harry Potter), where the initial acceptance letter is delivered by the regular Muggle post.
The very first letter Harry receives at Hogwarts is from Hagrid inviting him to tea at the end of his first week. Harry's family, OTOH, only tolerate his presence in their home, nothing more. They do send him something small at Christmas time… if they remember.
There is only one way that a student is selected to go to Hogwarts, and that choice isn't made by a person. In fact, it is made by a book and quill. In a tower at Hogwarts, where no student goes, sits the Book of Admittance and Quill of Acceptance. These objects decide who gets a place at Hogwarts.
On Harry Potter's 11th birthday he begins to receive letters inviting him to enroll at the Hogwarts School of Witchcraft and Wizardry. Each is addressed in green ink to “Mr H. Potter, the Cupboard under the Stairs, 4, Privet Drive, Little Whinging, Surrey.”
This allowed Hogwarts letters to be sent to all magical children at the appropriate time, even those who were Muggle-born. The Hogwarts letter consisted of a note of acceptance from the Deputy Headmaster or Headmistress of Hogwarts.
There are a couple of reasons why Mafalda Hopkirk mattered. Firstly, she was the one who wrote to Harry when he (or Dobby) performed illegal underage magic outside of school.
Ing to the rules of the Harry Potter universe, only those with magical ability that are born in the United Kingdom and Ireland (or move there at a young age and manifest magic that then gets registered by the Hogwarts Book of Admittance and Quill of Acceptance) will get a Hogwarts letter.
Wizarding post was delivered using owls. Some wizards and witches preferred to use their own personal owls, but the Owl Post Office was also a large and reliable network for owl delivery that offered speedy and safe communication.