As a British citizen you do not need a work permit to legally work in Spain. However, it is advisable that you apply for a residence permit as you will be there for quite some time and the residence permit will give you access to certain benefits.
If you are looking for a paid internship you might struggle as unemployment in really high at the moment in Spain. Getting a job offer will really depend on your level of Spanish.
If you are willing to work unpaid then you could try to approach companies directly and highlight what you can offer them.
There are also lots of companies that organise internships for students to meet their specific requirements. These comapnies will charge you fee but will also help you in terms of finding accommodation and settling into expat life in Barcelona. Two such companies that can offer you a tailormade internship experience are Connect-123 and IES Abroad.
For some general information on working in Spain check out this page on Working in Spain.