Before the Renters' Rights Act, most tenancy agreements included blanket "no pets" clauses, and landlords could refuse without giving any reason. The Act changes this by giving you a statutory right to request a pet and requiring the landlord to consider your request fairly.
When you make a pet request, your landlord normally has four weeks to respond in writing. They can refuse, but only if they have a reasonable ground. What counts as "reasonable" will depend on the circumstances, but examples might include the property being too small for the type of animal, a lease or freeholder restriction that the landlord genuinely can't override, or a valid concern about the specific type of pet.
Any existing "no pets" clause in your tenancy agreement is overridden by the Act. You can make a pet request even if your agreement says no pets. Your landlord can require you to obtain pet insurance to cover any potential damage, but they can't charge you a higher security deposit or an additional fee, which would be a prohibited payment.