Can somebody help me normalise this database/check my workings
You need a one-to-many table to list the types of pets that a person has, like

create table people_pet_types (
  person_name varchar(...) references people(name),
  pet_type varchar(...),

  primary key (person_name, pet_type)
);


Assuming name is the primary key of People (which it shouldn't be, else you can't have two people with the same name).

This allows a person to have many pets types.

Also, "Age" and "Alive" should probably be derived from date_of_birth and date_of_death (nullable). Otherwise, you need to update the table to re-calculate the age at least every year. You can use a VIEW to calculate the age, and alive values for you.