In the majority of instances, planning permission is not required to install a louvered pergola at the rear of a residential property, as these fall under permitted development rights. There are some exceptions however, including where your property is located in a conservation, green belt or AONB (area of outstanding natural beauty), your property is listed and where permitted development rights have been removed (usually in historic locations to protect the character of an area).
Additionally there are certain stipulations on sizing for pergolas under permitted development, which vary dependent on whether the structure is freestanding or lean-to, attached to the property, which limit the height to 2.5m or 3m respectively, when within 2m of the boundary. Furthermore the structure should not take up more than 50% of the overall outdoor space (including all other structures such as garden buildings).
As every local authority regulations vary, there is no one rule for all, so if you are unsure it is always recommended to check with the local authority.