Barrio Gótico is the deep Barcelona, the old side of the city, the narrow streets, the old buildings, the place where public buildings are... the heart of Barcelona...
Eixample means in catalan "extension". Its name comes from the 19th century, when the industrial revolution.
Easy to describe: a touch of bohemian, cosy restaurants and cafes, nice stores, small streets... ideal to have a walk and spending time with friends at night having some drinks!
Everybody has heard at least once this name: Gaudí. He is the genius of the modern art, the creator of many buildings and interesting places in Barcelona.