House, and therefore extension, looks to be post 1948, so permitted development rights probably ‘used up’. Under permitted development you could only do a single storey if extending to the side anyway. Therefore you will need to apply for planning permission. Looking at that photo it seems unlikely that you will run into too many problems if you aim to just extend the terrace, matching the existing scale and finishes.
Your local council website will have information on how to apply, and for a fee you should be able to organise a pre-application meeting with them to get a steer on what is likely to be acceptable.
EDIT: Misread the OP and thought you wanted two storeys. Single storey might be possible under permitted development, although I would think it’s unlikely, but worth double checking.